There have been some really strange characters in video games throughout the years, but some that have always stood out to me as really weird are:

Earthworm Jim: He's a man with a worm for head that fights bad guys using laser guns while wearing a white jumpsuit. A very strange character

Boogerman: A caped crusader, whose powers are to throw boogers at people. Need I say more? He was always a super strange character, and to this day I have no idea what the developers were thinking when they made him.

Zappa: This guy looks like he crawled straight out of the film "The Ring", had his back broken in the process, and walks in the reverse crab stance. Not to mention that when he fights, he actually summons a girl who looks exactly like the girl from "The Ring". He's such a weird looking character, but Guilty Gear is known for their strange characters, that's why I love the games so much!

The Blob: Clayfighter has a lot of weird characters, but The Blob was one of the weirdest. He was just an amorphous blob of clay. He could shapeshift and fight in a variety of forms though.

Blue Suede Goo: Another Clayfighter character, this guy was supposed to be a parody of Elvis, and he is just really weird. His appearance, his mannerisms, everything is awkward about this guy.

Tingle: Tingle from Zelda is scary looking, I've always thought he looks like some kind of pedophile or something. He's a very strange looking character, and his appearance in a Zelda game always creeps me out.

Voldo: Soul Calibur's Voldo is the exact same as Zappa to me, a really creepy looking guy who walks like his back has been broken several times.

Vectorman: Vectorman was a great game, I loved it. However, Vectorman himself was always weird, to me. He looked like he stepped out of the old school fighting game "Ballz". A character made entirely out of spheres. His games ruled though.

Agent York from Deadly Premonition. I love that strange motherfucker.

RIP EWJ and Vector man, its a sad SAD day without new games from you two.

Klayman from Skullmokeys

Hell yeah! Green robot (made of orbs) sent to clean up the polluted Earth and thwart off evil! (Reminds me of Wall-E...)


Also, Banjo-Kazooie is rather odd. An anthropomorphic bear in yellow shorts, with a blue backpack, and his wise-cracking, ill-tempered bird friend.

Edit: Toejam (& Earl)!

Now that would be weird if Banjo wasn't anthropomorphic like the most animal protagonists are whether it's about video games or cartoons.
More "realistic" animal fighting game: Overgrowth

How about the Prince from Katamari Damacy.

Nobi Nobi Boy, by far.

Every character from Loaded (deranged psycho paths but compared the rest of the galaxy they seem to be the only psychopaths that can be labelled "good guys"
Look up Fwank if you're curious heh.


Gex

Abe

And of course Earthworm Jim 8D!

Chell.

You can only stay quiet for so long.

Nah, Abe wins that title.

Parappa the Rapper

Sir Daniel Fortesque

Guy from Altered Beast

Can't believe no one mentioned Lester the Unlikely. Seems very unlikely.

Duke Nukem.

My favorite strange protagonists:

CAPTAIN RAINBOW


MISTER MOSQUITO


SIR DANIEL FORTESQUE

Link. Who else walks into your house, breaks all your shit, takes your money, then rolls into your door on the way out?

Alcatraz from Crysis 2, doesn't even talk in the whole game.

You are right my friend.



Also, the Old Lady from Dementia.

Do you know Aqua Teen Hunger Force? Our beloved game developer from the past, Midway Games, had made one of the strangest games of mankind (simply because it's a license game of ATHF). That is Aqua Teen Hunger Force Zombie Ninja Pro-Am, filled with strange characters with rather special personalities and with a wacky name given by Midway.

Click the picture for a trailer. As for who is the strangest of them, I'd choose the whole happy meal in this case (Frylock, Master Shake & Meatwad).

Torque from The Suffering. (Depending on his morality)

I’m going to go with Kiddy Kong. From Donkey Kong Country 3: Dixie Kong's Double Trouble! Simply because he’s a baby and I don’t know anyone who likes him.