I think in terms of a storyline, established characters can have a backstory that kind of correlates with the original storylines of the earlier plots of MK.
So in terms of each characters' backstory and motivations in Mortal Kombat it be plausible to explain them in terms of each character breakdown. So...

BO RAI CHO:
In the earlier stages of his teachings to Liu Kang from the White Lotus Society, was out of shape and drunk in slumber. Although Liu Kang learned very much from Master Bo Rai Cho, the knowledge that he possessed, he couldn’t apply due to his out-shaped figure and lack of daily practice. So instead, Liu Kang was chosen over Master Bo Rai Cho to compete in Mortal Kombat. Later on, Bo Rai Cho witnessed the destruction of his brothers at the hands of Shang Tsung and decided to accompany Liu Kang on a latter tournament held under the rule of Quan Chi and Shang Tsung. Unfortunately, Master Bo Rai Cho, was nothing short of an embarrassment in the tournament and was kicked out in his first bout at the hands of Drahmin. Feeling ashamed because of his title, but yet determined to leave a mark to be felt by both Shang Tsung and Quan Chi, Master Bo Rai Cho delved into a period of reflection and reinvention of himself. He spent several years retraining himself in the rigorous teachings of kung fu styles that he learned so many years ago. Bo Rai Cho enters Mortal Kombat now as a renewed spirit with a motive of redemption to the memory of his fallen Shaolin Brothers from the White Lotus Society

KAI:
Born of half Chinese half African-American heritage, Kai was brought to the White Lotus Society as an orphan. It was there that found refuge and was taught the discipline of kung fu martial arts training. For years Kai studied under the direction of Liu Kang and Bo Rai Cho at an early age and developed his style of fighting from his treacherous and intense training. Years would past, and Kai would have completed his training with the White Lotus Society and mastered the spirit of the “Tiger”. Elder statesmen and monks of the temple, instructed Kai, that in order to learn the top techniques of the White Lotus Society he would have to possess the spirit of the “Dragon” and be mindful of the harm and extents of his powers can have on weaker people, in that he would have to find it for himself outside the temple. Years would past and a raid by Shang Tsung’s Outworld forces would destroy the White Lotus Temple shortly after Liu Kang was selected as the sole representative of the Temple to fight in Mortal Kombat. Liu Kang would leave the temple and fight for the Earthrealm in Mortal Kombat due to the destruction of their temple while Kai had left the Temple on a journey of enlightenment. Once Kai returned to the remnants of the temple, he was enraged with anger at what laid before him. Kai would ally with Liu Kang in their journey into Outworld to fight against Shang Tsung. After Liu Kang won the Mortal Kombat Tournament, Kai was infuriated that he chose not to kill Shang Tsung in the tournament. Taking matters into his own hands now, Kai followed the defeated Shang Tsung back to Shao Kahn’s kingdom against the wishes of Liu Kang. It was there that Kai would meet Shao Kahn face to face and be quickly overmatched. Kai loosing his fight with Shao Kahn was taken and enslaved for his misguided anger and was entranced under Shao Kahn’s mythical powers to do Kahn’s biddings as an assassin. It would be later revealed that Kai would be recruited by the renewed Shang Tsung as one of his elite assassins for the next Mortal Kombat tournament.

FROST:
Frost, being ostracized most of her life for being an inexperienced fighter in the Shirai Ryu, was eager to develop into a renown fighter to prove everyone wrong and gain their respect. Shunned from any battle training from the Shirai Ryu, she faked her death and set off in the night from the walls of the Shirai Ryu for a journey of enlightenment. Several years past and after several sparring sessions with local masters, she set her sights westward and encountered Sub-Zero of the Lin Kuei in civilian clothes. In her encounter, Frost explained her will to find the ultimate fighter, and decided to spar with Sub-Zero to gage his combat skills. Apprehensive at first, Sub-Zero refused to fight, until he was forced to out of Frost’s sheer desire. As they engaged in combat, Sub-Zero saw potential in her skill but was still unimpressed. After their sparring session he left to this dismay of Frost. Frost being more impressed with his skill decided to follow him back to the sacred Lin Kuei Temple and snuck behind their walls with her novice spying skills she learned. Eventually, Frost got caught and was brought to the Abbot and elder officials of Lin Kuei and Sub-Zero. The Abbot and elder statesmen of the Lin Kuei had a strict policy of turning away outsiders from the Temple, with a penalty of execution. Sub-Zero reminiscent of their spar days earlier showed compassion and decided to take her up against the policies of the Lin Kuei. It was decided that only through her submissiveness to the culture of Lin Kuei and guidance of Sub-Zero’s teaching would her life be spared. Sub-Zero that day was held responsible for teaching Frost, Lin Kuei culture as his student. Being Sub-Zero's student in the Lin Kuei, Frost trained very hard under Sub-Zero but became quickly frustrated with her progress. Her relentlessness and impatient nature caused her to abruptly end her training with Lin Kuei after novice level teachings. So one night within the Lin Kuei Temple, Frost, knowing with expert precision when the upper chambers of the Lin Kuei Temple was unguarded stole the sacred Ei Ching manual and disappeared into the night not to return to the Lin Kuei in years despite several attempts by the Lin Kuei to track her down. When Sub-Zero learned of her alliance with Shirai Ryu, Sub-Zero ordered an all out war against the Shirai Ryu some time after causing the heated rivalry between the two clans. Frost reunites with her Sifu in Mortal Kombat as a more powerful and skilled fighter, with Sub-Zero's vengeance against her and a mission to free her ninja clan members Black Widow, Forage and Poison Ivy from Shang Tsung's control.

Frost in her power hunger quest to be become the best fighter, steals the Ei Ching Manual of the Lin Kuei and disappears from the backlash of the Lin Kuei and Sub-Zero. This action only fuels Sub-Zero distrust to take on new students. Frost then takes the manual to a remote area in the Shirai Ryu to learn and train herself the techniques of the Lin Kuei and ice control techniques unbeknownst to Scorpion, Black Widow, Forage and Poison Ivy. A few years past, and Frost's fighting skill level develops to be one of the best out of the Shirai Ryu to the surprise of everyone in her clan, including Scorpion. A special twists occurs though to where the Lin Kuei send a spy to map out the Shirai Ryu's village and spots Frost. The spy returns to Sub-Zero and the Lin Kuei to report the discovery and as a result orders a surprise raid on the Shirai Ryu. During the raid, Scorpion gets killed and Black Widow, Poison Ivy and Forage and a few others from the clan escape including Frost, and the Ei Ching Manual from the Lin Kuel ninja clan gets destroyed and burned to ashes in the war to never be recovered. During the raid, Shang Tsung took on the image of Sub-Zero and turned out to be Scorpion's murderer so witnesses and Scorpions specter place the blame of his murder on Sub-Zero when in fact it was Shang Tsung who commits the murder. Years later there is another raid this time from Shang Tsung's band of warriors on the remnants of the Shirai Ryu, to where he takes Black Widow, Poison Ivy and Forage hostage for recruitment of his legion of elite assassins to protect him called the Divine Nine. He enslaves Black Widow, Poison Ivy and Forage under his sorcery and Frost is the only one left behind. So she as the one of the few representatives of her clan left with a guilty conscious decides to enter the tournament to free her sisters of the clan and take revenge for the raid.

This can also create a feud between Sub-Zero and Frost due to the fact that he feels he stole the prized possessions of the Lin Kuei and some of their most sacred fighting techniques and ice abilities. So Frost encounters a love/hate relationship between her and Sub-Zero though she still regards Sub-Zero as her master for teaching her how to fight but a hatred for the massacre of her clan. She also has a fuel raged hatred for Shang Tsung for the capture of her sisters Black Widow, Poison Ivy and Forage and feels she's there only hope to survive and be released from his control.

FU JIN:
Once Raiden learned of the threat of Earthrealm and Outworld merging into one, he went to the elder gods to seek help. Once in their presence, Raiden immediately explained the situation. To his surprise, he was sent Genesis, a trusted guardian of light, from Luminous the god of light. He also received aide from Fu Jin himself, the god of wind. Known for his calm and introspective nature, Fu Jin weighed the pros and cons of the situation and decided to help. Due to the rules and regulations of the Mortal Kombat tournament, Fu Jin then took upon the image of a mortal being and descend down with Raiden and Genesis to combat the merging realms. Aware of the magnitude of the described situation from Raiden, Fu Jin was taking a big risk of his godhood to compete in this tournament. Loosing the battle of the merging realms meant being stripped down to a mere mortal. Fu Jin allies with Raiden in Mortal Kombat with his godhood on the line and the threat of Earthrealm being destroyed by Outworld.

RYU (OF STREET FIGHTER) If selected as a guest/bonus character:
Noted for his reputation of being a consecutive Street Fighter Champion Title Holder, Ryu attracted attention from random underground street scufflers. Being used to the confrontations, Ryu would often walk away from the encounters in respects to avoid bringing more unwanted attention to himself and loosing face as Titled Champion. Only in the extreme circumstances where he'd be forced to defend himself would he lash out in pinned up anger and fury against contenders, which he has proven to do a few times. One untypical day while minding his own business, he was called out by a unknown fighter parallel to his to skill set from across the streets in defaming his character and the honor of the Street Fighter Tournament to a point of being provoked to defend himself and the title of Tournament Champion. With this random act of violence, Ryu engaged in a one on one fight with the fighter. Loosing his focus and cool, Ryu valiantly fought by eventually was overcomed by the unknown warrior in the surprise encounter. This would cause the fighter to become more patronizing and eventually dismiss the relevance of the Street Fighter Tournament all together. Broken and bruised, Ryu, would pick himself back up after the thrashing and seeking meaning behind the experience with reconditioned training from his Sensei. He would also use the encounter to investigate the unknown warrior's whereabouts after a few days. The few days investigation would lead him into a mass of swirling clouds in the sky and a whole new realm in which a famed tournament where Earthrealm and Outworld's finest warriors would be competing in a event called Mortal Kombat. A sudden flash back of disrespect this unknown warrior would profess towards the Street Fighter Tournament would fill Ryu with rage. Finally, with a mind set on tracking down this unknown fighter, Ryu, walked aggressively through the spiraling clouds and enters Mortal Kombat to restore honor to his loss of reputation and honor of the Street Fighter Tournament.

SHINNOK (as boss of next MK):
After it was communicated that Shao Kahn lost control of the tournament to the hands of Raiden, and the realignment of power and realms where to shift under his authority, Shinnok seized what to him was a golden opportunity. He would travel from the NetherRealm into Earthrealm with his Black Magic sorcery and form an alliance with the Outworld Emperor, with him and Shang Tsung serving as subservient underbosses to his strategy and plans to regain control of the Order of the Realms amulet and consequenetly power over the existing realms in question. The defeated Shao Kahn desperate for power, agreed to the terms and handed over his dominion in Outworld to Shinnok in hopes to gain access to Earthrealm again. Now with the powers to be resurging again for access Earthrealm, Shinnok now takes the lead in NetherRealm and Outworld's ambition to rule Earthrealm in the tournament Mortal Kombat.

TANYA a.k.a. ALIBI:
Alibi, unlike many others, came into the Shirai Ryu at a later age. She was looked upon as a late bloomer by many in the clan. So by the time she had gotten to the age of fighting, her skills weren’t as keen and honed as the others. To make up for weakness in physical ability, she had a mastery of spoken language. She could speak seven different languages (including English, Spanish, French, Japanese, Italian, Russian, and Swahili). She also possessed a well-keened skill of persuasion, and possessed a born psychic ability to read and control people’s minds and because of this skill she was given the initial code name of “Alias“. Although in other circles, these skills might have been looked upon as great, the Shirai Ryu saw her skills as inferior. She often would stay behind on missions and be restricted from learning any advanced fighting techniques. But eventually she would gain the trust and respect of the heralded Scorpion and became his student. Being older than his usual students, Scorpion had to demonstrate patience with her, a virtue he wasn’t use to exhibiting as a Sensei.

The initial phases of their training were routine until the Lin Kuei decided to abruptly end their training by waging a war. Being at novice level of fighting she fought well in the war but witnessed the death of her Sensei. Tormented she fled to a nearby town until the war receded. Months later she would be visited by Scorpion’s specter in random visions and dreams in her sleep. These visions drove Alibi to fight in the Mortal Kombat 4 tournament where ironically enough, part of her identity would be compromised. Tanya, as it was later revealed, had her focus affected with the leak in personal information. The leak in information would cause her mind to wander about other things instead of focusing on her opponent, causing an early round defeat in the tournament. She would also reunite with her former Sensei, Scorpion in the tournament. Scorpion saw that she competed well in the tournament and respected her efforts but scolded her to her face. Feeling that her skill was average at best, he demanded that she train harder, and mediocrity was not acceptable as a member of the Shirai Ryu. So with the events unfolding, and Shirai Ryu now extinct, she went into an isolated location to train with the resurrected Scorpion to instruct her on occasion. This time in her life represented a period of purification from her former self. She was given the new code name “Alibi” by Scorpion himself and honed her psychic abilities into a useful fighting weapon against foes. Alibi enters Mortal Kombat for redemption and for her resurrected Sensei.

ALLURE (new character proposal):

Ever since the earlier part of her life, Allure, garnered attention and affection from everyone she surrounded herself with, especially men. She was known as the most popular girl in school and hung out with only the most popular people within her inner circle. Eventually because of her adored beauty and intelligence she would move out to California where she’d chase a modeling/acting career in the field of cinema. To keep herself in shape, she would exercise and partake in kickboxing, Tai Chi and yoga to meet the demands of a Hollywood image. Her initial time in California produced little to no gigs, having to work small odd and part time jobs. Her luck would change and she would eventually be introduced to some powerful media giants within the region and build successful relationships with her connects, elevating her career to more lucrative gigs and positions. And because she naturally had “the look” and intelligence, she fit right in to the culture of Hollywood. She would meet various celebrities and powerful moguls in the industry with her charming persona making her an eventual celebrity. One day, after being hounded by paparazzi, she would be approached by a high level mogul to play a lead role in an upcoming movie. They would arrange a meeting for her to read a copy of her script and attend an audition for the part in a state of the art studio with the producer and casting agents for the movie. Allure would show up to the audition early eager to land her first lead role until she would be engulfed into a magical portal into Outworld. Allure enters Mortal Kombat by deception and must fight her way out of the realm Outworld to save herself.

ARCTICA (new character proposal):
Arctica came into the Lin Kuei as an abandoned child along the same time that Sub-Zero came into to the ranks of the Lin Kuei. Together throughout the years they grew as renown assassins within the temple and studied under the same teachers. They progressed through their training around the same speed and built a bond of friendship and professionalism amongst each other. This was until Frost infiltrated the Lin Kuei. Frost’s presence was polarizing and divisive. Being perceived as an outsider, Arctica did not accept Frost and as a result her trust and perception of Sub-Zero changed for the worst. Sub-Zero’s compassion and mercy for Frost’s life, caused her to look at him as soft and weak. Through Frost’s training, Sub-Zero’s dissention grew into a full blown enemy in the eyes of Arctica, whom now took the stance of Sub-Zero’s counterpart. Through premonition, Artica foreshadowed a war between the Lin Kuei and rival clan the Shirai Ryu to ranking members because of Frost’s presence but her outbursts fell on deaf ears. Her later premonition proved to be right. As a result of Frost’s disappearance with the Ei Ching manual the actions incited a conflict between the Lin Kuei and the Shirai Ryu. Arctica was sent out by the Lin Kuei to map out the Shirai Ryu’s grounds as a spy and came back to report where the Lin Kuei artifact was being held. Once discovered, a war between the two clans ensued resulting in the massacre of the Shirai Ryu and the destruction of the Ei Ching manual. In response Arctica was given the mission of assassinating Sub-Zero for his act of treason. The loss of sacred artifacts of Lin Kuei’s history, culture, and secrets caused them to integrate an automation process with se lect warriors within its ranks that showed unfavorable conflicts with the ideals and philosophies upon which Lin Kuei was founded upon. So Arctica, having the same Black Arts training of manipulating ice as Sub-Zero, now enters Mortal Kombat as a hired assassin against Sub-Zero for his act of treason against the Lin Kuei shortly after his banishing from the temple.

RED VIRTUE (new character proposal):
Long time standing member of the Lin Kuei, Red Virtue, was inducted into the temple as an outsider with a mysterious but dark past. Physically attractive and beautiful, Red Virtue always had garnered unwanted attention from various men that would eventually motivate her inception into the Lin Kuei. And being of a foreign background, Red Virtue had to pay a series of long and intense dues to be inducted. It was because of these experiences within her faint past and completion of intense Lin Kuei ceremonies that she was to be considered. The combination of paid dues and her dark and painful past caused Red Virtue to developed a strict, intolerant and emotionless persona towards others within the temple. She would become an instructor for Kunoichi Ninjas within the Lin Kuei. Her leadership style is militarized, strict and intense exhibiting no remorse or tolerance for emotions, mercy and weakness. With her decorated tenure within the Lin Kuei she commands a mid-level authority over female assassins. When word of an outsider infusing a war between the Lin Kuei and the Shirai Ryu came to her attention, she sent long-time advanced associate “Arctica” to investigate. Receiving the inside information from her, and learning of Sub-Zero being the root cause made her and other high ranked Lin Kuei officials to take a decisive action to eliminate him. Now as Artica enters Mortal Kombat so does Red Virtue to uphold the honor and reputation of the Lin Kuei.

OBSIDIAN BLUE (new character proposal):
Little is known about Obsidian Blue’s origins. When questioned about her background, the information that has turned up has proven to be extremely vague for she doesn’t have ties to any martial arts schools, clans or teachers. She has been observed to be a very discreet yet powerful warrior with mysterious motives to all. With mixed reports of her past, to some she has demonstrated herself to be a heroine to niche communities, providing support and justice to aide in the pitfalls of selected areas. And to others she has demonstrated herself to be a menacing villain inciting riots and chaos from her short hinged presence in other areas of the world. In receiving word of a world renown tournament consisting of the best fighters on Earth and beyond to participate, Obsidian Blue enters Mortal Kombat as a wildcard and mystery to all that have heard of her legend.

OBLIQUE (new character proposal):
A model citizen from the planet of Utopia, Oblique was heralded as a celebrity in her native providence until one untypical day, her planet was invaded by Shao Kahn and his army of Outworld warriors. The attack came to much a surprise to her native people in that their counter attacks were sporadic and unorganized, and within a few weeks her entire planet was conquered by Shao Kahn superior soldiers and weaponry. Because of her beauty, Shao Kahn decided to take her back to Outworld where she was repeatedly tortured and abused by Shao Kahn and other highly ranked members of Shao Kahn’s circle. After years of torment, her biological makeup gradually began to break down into a liquid molecular structure giving her the ability to transform into liquid masses at will. This allowed her to escape her cell in the lower chambers of Kahn’s dungeon into a desolate area in Outworld where she would meet a mysterious monk escapee from Kahn’s prisons. Old and decrepit the monk saw Oblique’s condition and decided to nurse her back to health through martial arts training, exercise and field work. The physical activity allowed her to gain her strength back and taught her to control her liquid techniques and transformations. Years later, the monk dies leaving Oblique in isolation and nothing but the memory of what Shao Kahn did to her home of Utopia. Oblique enters Mortal Kombat with no alliances to anyone and a heart filled with vengeance against Shao Kahn and anyone that gets in her way.