I have quite a few characters waiting to be used in a future campaign and a few of them fit this thread's criteria.

A racist human supremacist wizard insulted the wrong person, a witch who cursed him so that he would transform into a random race (never human) each night during his sleep. His racial traits change accordingly each time and his appearance is chosen by the DM. I would ask the DM to make NPCs make racist remarks toward him on a constant basis (e.g. as a halfling, an NPC may make fun of his small stature), but also occasionally turn this into an advantage (e.g. as a orc, he can pretend that his companions are prisoners or slaves and pass by a group of orcs without fighting). The curse also allows him to magically understand his new race's language(s) so that he can communicate and learn from his "kind" that other races aren't inferior. The curse is unbreakable, so spells have no effect on it. He doesn't know it, but falling in love with a woman from a different race will make him permanently transform into that race, thus "breaking" the curse. The DM may choose a different conclusion to his story, but it's obvious that it's based on the Beauty and the Beast. The falling in love with a different race is just a suggestion in case the DM has no idea what to do with it.

An orc berserker barbarian with proficiency in performance and a dulcimer. He was born with an anomaly that makes him extremely empathetic, so much so that it's debilitating. He starts crying as soon as he sees other people suffering. The only exception is when he goes into a rage, but once it stops he cries after seeing what he did. This obviously made other orcs in his tribe make fun of him, but he survived because he was strong, really strong. Other orcs learned not to mess with him, but they still made fun of him in his back. He was also protected by the chieftain, who was his father. One day during a coup, his father was assassinated and the son was expelled from the tribe. He travelled the world, giving him the outlander background. Among his travels, he met a blind female human bard. She taught him how to sing and play the dulcimer, not knowing that she had befriended an orc. They both fell for each other, but upon touching his face she had a shocked expression on her face and he hurriedly left and never came back. Whether or not she's brought up again in the story is left to the DM. I would ask the DM to present him to the party in a way that would surprise them, e.g. he's presented as a potential hostile NPC to the party, until it's revealed that he's in fact a PC and that I'm the one controlling him in battle once his turn is up. I exploit his Powerful Build feat by making him carry tons of hunting traps that he loves placing everywhere. "Before I leave this dungeon room, I place a hunting trap in the doorway. I also place twenty of them all around our campsite. Oh, and I place one here, here, here, here, and here, oh and here, and here, and here..." Whoever said that half-orcs are better than orcs don't know how broken their racial feat can be.

A lawful good inquisitive rogue with the investigator background. He's basically a walking lie detector with a guaranteed minimum of 17 on all insight rolls for detecting lies. He also has a Strength of 4 because I rolled for stats and that's what I got. He has expertise on insight and investigation, and proficiency on acrobatics, medicine, perception, sleight of hand and stealth. To explain his weak strength, he had both of his arms broken when fighting against a criminal. He healed but his bones are still weak and he can't carry a lot of weight. It's so bad that even carrying an extra potion causes encumbrance. Inventory management becomes one hell of a challenge. He keeps manacles on him so that he can arrest criminals. He's the type of man to stabilize and save an enemy from dying so that he can put him under arrest and allow him to receive a fair trial. This also explains the proficiency in medicine, he's not strong enough to carry many healing kits on him. Plus it can be useful to determine what illness or poison killed a potential murder victim, he's the type to do autopsies and perform first aid on both innocent victims and neutralized criminals.

I have more, but this will have to do for now.

So, I described a teifling sune monk a while back who had memory loss after waking up naked and beat up in a tavern. Well, turns out she was a true polymorphed chair all along. (Why is this a common theme?!?!?) and a perv wizard just wanted to see a naked teifling at the time, so he polymorphed a chair into just that. He just forgot to change her back... (my dm is on drugs, I swear.)

I have a couple more ideas floating about for characters I might want to play one day.

One is a bugbear Wild sorceror who was Once a Man (tm) until he got caught in a portal to Limbo and returned changed. He's a Noble, though, and he still acts like one and wears fancy clothes and stuff but he's a ,y'know, bugbear.

There's also a rogue who's so good at Sleight of Hand, Deception and Performance that they're able to convince everyone that they're a powerful caster. No actual spells, just a person who's an extremely good stage magician.

Nice.

Kick them out. 

Just finished playing (he died) a warforged bear totem barbarian.
To get more into being a bear totem barb, he had skinned bears and wore their fur glued to his body but also wore a very badly made taxidermy bears head.
Also he would converse with bears who just seemed to accept him as one of their species - led to an amusing conversation between my char and a wandering bear.

Quote from Sly_Fox >>

Kick them out. 

 I'd agree with you, except that that seems to be a bot that's copy-pasting someone else's post with an inserted spam link.

A Kobold heavy weapon fighting rune knight, with a focus on grappling. He is both cute and scary

An orc pacifist ranger with the grappler feat, this is my friends character, not mine. The orc is a ranger to make them more useful in a fight because the orc will not kill, only paralyze, knock unconscious, or neutralize foes in some other way. Also, just to be funny, the orcs name is Yokgol The Butcher

Quote from Cheeky_Hamster >>

A Kobold heavy weapon fighting rune knight, with a focus on grappling. He is both cute and scary

But why? He should always have disadvantage on his attack rolls and most of his grapples. Anyway, that reminds me of a player in my old group. He made a kobold as a side character and presented it to the main party in their newly owned tavern. We didn't get to use him because the DM disbanded the group shortly after. I don't remember all the details, but the kobold told us he had met a celestial, I think it was a unicorn, and he formed a pact with it. A celestial warlock kobold is quite strange, but your example is much stranger. He was also kind of a wimp and easy to startle.

Just... an incredibly broken homebrew when I first started playing D&D and didn't know any better. He was a lionfolk rider, meaning he had a mount that attacked with him... which was an owlbear. Who had multiattack. So he was virtually invincible and did so much damage it was ridiculous. Not sure if this counts since it was just SO broken, but it was really fun! Luckily none of the people in the party knew much about D&D so it didn't ruin the experience for anyone; we were all running pretty broken builds. I could never play him again, but I do still love the image of a big lion man in full armor atop an owlbear.

Beautiful. Just. Beautiful.

Quote from Flaxo >>

Quote from Cheeky_Hamster >>

A Kobold heavy weapon fighting rune knight, with a focus on grappling. He is both cute and scary

But why? He should always have disadvantage on his attack rolls and most of his grapples. Anyway, that reminds me of a player in my old group. He made a kobold as a side character and presented it to the main party in their newly owned tavern. We didn't get to use him because the DM disbanded the group shortly after. I don't remember all the details, but the kobold told us he had met a celestial, I think it was a unicorn, and he formed a pact with it. A celestial warlock kobold is quite strange, but your example is much stranger. He was also kind of a wimp and easy to startle.

Whilst obviously not the strongest character there is a surprising amount of synergy

In direct sunlight Kobolds have disadvantage anyway, so using a heavy weapon is no problem, as it can’t stack, then if you attack near an ally pack tactics makes it a straight roll.

Rune Knight allows a character to grow to large size 2/day meaning the heavy weapon isn’t a problem at all inside or out of direct sunlight for the important fights. And it’s so much funnier and cooler to go from small to large, than from medium to large.

Rune Knight is also amazing for grappling and it’s really funny to see something so small be so effective at it (which if you build right is quite easy). Also grappling, you don’t need to be able to see, no disadvantage therefore on the contested Athletics check.

Hardest part is actually just the -2 to strength, which as he is a giant themed character just means that I’ll focus on trying to get [magicitem]Gauntlets of Ogre Power[magicitem] (or Giant’s belts). But I also rolled a 17 so a 15 is not a bad start.

Oh, I didn't know about those rules concerning rune knights and kobolds. It does make for an interesting character, I give you that. Anyway, I've recently had an idea for a new character who fit this thread's criteria. His backstory is that he has retrograde amnesia (read what follows before you say it's unoriginal). He doesn't remember his name or anything about him, but his semantic memory (i.e. knowledge about the world, language, etc.) is unaffected. After some time, it eventually becomes apparent that he has special powers, but he ignores the origin of said powers. What he doesn't know is that he's a warlock who made a pact with a devil. Another thing he ignores is that his alignment changed, he used to be lawful evil and now he's chaotic good. Whatever happened in his past that made him lawful evil was now forgotten and no longer affecting him, thus resetting his personality to chaotic good. He constantly feels the urge to do the right thing no matter what the law says. When the innocent is in trouble, he simply has to do something to help. I'd let the DM create the rest of his backstory, so this character is meant for creative DMs who are great at world building. Whatever he did in the past, who his friends and enemies were, his real name, what caused his amnesia, why he made a pact, what his motivations were, his family; all of that is left to the DM's discretion.

I have just created a charming rogue in a homebrew campaign, and uses cuteness as a way to get what she wants. She'd cover herself in blankets, put on adorable outfits, act as innocent as possible, then steal the party's life savings once they trust her. And because she's an adolescent tabaxi, she's just the cutest little thing.

Since you said bizarre I feel like I should chime in I played a cavalier fighter named Jonny Star who couldn't walk and instead rode a horse to get around loved him

Beautiful

Awesome.

My first character, a tiefling rouge who was exceptional at stealth and slight of hand and almost never failed. A common thief who knocked people unconscious with essence of ether and would take everything off them, including clothes.

also one of my friends character is a LE halfling wizard who summoned asmodeus.