Just to add something extra for the OP to agonize over (
), the "crazy homeless guy" is often considered an overused negative stereotype in its own right. To be honest, I'm not sure you have to worry
too much about diversity with such a character - both the homeless and the mentally ill are pretty marginalized members of society already. Were I in your position, I would be more concerned with portraying the character in a nuanced and sympathetic way while steering away the old comic stand-by of the insane, cackling bum guzzling down cheap hooch and marinating in his own filth (not saying
you are guilty of this, of course, just that its a pretty pervasive image society has of the homeless).
I can understand your dilemma, though. Certain ethnic groups sadly
do fall victim to being stereotyped as homeless addicts, but when you find yourself that sort of "rock and a hard place" situation, your best bet is, IMO, just to accept that you will probably be criticized regardless of what you write, and go with your gut. White or black, gay or straight, christian or atheist, focus on making your character and those with whom he (she?) interacts feel like real, sympathetic, flawed human beings in a setting that rings true.
How many other characters are in this story? Are they all mostly other homeless people, or more well-off. If you're worried that, say, an African-American homeless man could be taken the wrong way, then they could always have a unpleasant altercation with a rich, entitled snob who also happens to be African-American. Depending on where the story set, the neighborhood could be almost exclusively one ethnicity or another, or far more mixed - and in places that suffer from serious racial divides, its not unusual for the well-off to be primarily of one ethnicity and those in poverty to heavily comprised of another. Some research on the demographics of your chosen location could go some way to easing your concerns here.