Here's the basic idea. You play the new person to move in, be them male or female.
You live a mostly normal lifestyle. Sure, you have your issues, but most people do. You live happily in an apartment that you feel you got a good deal on. The landlord said something about no one renting it due to the guy next door (who the landlord let slip was some sort of charity case that he let stay there for free). But you have yet to see your neighbor, so you don't really know what all the fuss was about.
About four months into your stay at the complex, you finally meet this mysterious neighbor of yours. He busts into your apartment, trashes the place, and then proceeds to have a mental breakdown in your living room. You think you may know what the fuss was about now. As it turns out, the guy's a schizophrenic who's caretaker died several years ago in a car accident. He's been living by himself in a depression, and never leaves his apartment. He survives solely on the care packages that people leave outside his door out of pity.
However, once seeing the man in his broken, filthy, and neglected state, how can you just let him scurry off back to his room to continue suffering? You suffer a loss of common sense and offer the clearly deranged man help. Perhaps all he needs is someone to look out for him, right?
This has potential to be a romance RP, or just a platonic best friends RP.
Bonus points if your character has some kind of mental issue as well, such as OCD, anger management, bulimia, or whatever. Then the two can be freaks together.