Well the following isn't Goro's bio as written in the games, but it describes in more complete detail than the MK1 and MK4 bios do.
Goro, the 2000 year-old prince of the Shokan (a race of four-armed half human-dragons), was invited by Shang Tsung to compete in the Mortal Kombat tournament, after Tsung lost the title of Grand Champion to Kung Lao. Goro defeated Kung Lao, and reigned as champion of Mortal Kombat for 500 years. Then he was defeated by Liu Kang, a descendant of Kung Lao. He was not, however, killed by Kang. Shang Tsung, furious to see Goro defeated, went after Kang himself, while having Goro fight Johnny Cage, Kano, and Sonya. Before that battle finished, however, Liu Kang defeated Shang Tsung, and the island on which the tournament was held collapsed. At this point, Goro was presumed dead by his father King Gorbak, as well as the Outworld emperor Shao Kahn. However, Goro returned years later, during Shinnok's escape from the Netherealm, determined to take vengeance upon Liu Kang for defeating him. He also became ruler of his people. Before he had the opportunity to battle Liu Kang, however, he was befriended by the Edenian Princess Kitana, who persuaded Goro to sign a treaty for peace between the Shokan and their enemies, the Centaurs. After agreeing to the treaty, Goro confronted the warrior Kung Lao, a descendant, like Liu Kang, of the great Kung Lao Goro defeated over 500 years before. The two settled their differences. Meanwhile, Goro had bigger problems to attend to; he joined Kitana in her war against the weakened Outworld Emperor Shao Kahn. During the war, Goro was attacked by Noob Saibot, an enigmatic assassin sent by Shao Kahn, and in a furious battle, Goro was fatally wounded. After his death, he was put in a coffin and lowered by his loyal people into the molten lava of the Kuatan Palace.
That info is based on his MK1 bio, the official MK1 and 2 comic books, his MK4 bio, Kung Lao's MKGold ending, info from Kitana's Konquest and ending in MKDA, and Noob Saibot's ending in MKTE. I wrote it this way to make it as complete and continuous as possible.
