I gotta commend you for really wanting to do it well! I have a lotta admiration for not being afraid to ask and really taking the advice and recommendations to heart!!
I have a mentally ill villain as well! There's two things I try to keep in mind for this:
-- I try not to look at it as "will someone get offended," but more "is someone going to be hurt." I have a very close friend with Dissociative Identity Disorder/Multiple Personality Disorder, and when we see yet another character with a surprise evil alternate personality, it's not that we're OFFENDED.... it just sucks, because multiple people are so mistrusted for that reason already! When you see something that looks like what you've experienced, it's exciting -- then when it's just an excuse to make a character frightening, it... it just feels bad. I just don't wanna make anybody feel crappy in the way I portray my characters!
-- the second thing that can really help with this is to not make the mental illness the CAUSE for villainy, or a way to make the character scary. To bounce off my DID/MPD comment, the most positive portrayal I've ever seen of MPD came from a villain -- Good Cop/Bad Cop from The Lego Movie. He has two very different personalities that are working together and occasionally disagree.... but that quality is not what puts him on the bad guy's side. The mental illness can definitely affect the way the character acts and the things they do, but be careful of falling into "he did X bad thing because he's crazy" reasoning. I feel like even piling on sympathy makes it hard to mitigate that unless the mental illness is really the point of the story!
I'm not an expert so I can't give a specific recommendation, but a basic place to start would be just looking up some things on wikipedia, for the simple reason that it'll link you to other, related mental illnesses. So you look up things like Bipolar or Borderline, read the article and see if it might fit, and then start looking into it in-depth if it seems like it might be a good match -- or start clicking on other mental illnesses that might fall under similar symptoms or the same general banner if it doesn't quite work for your character.
A really invaluable source for research I'm gonna suggest is support forums for whatever mental illness you choose -- hearing about the experience straight from people who've actually dealt with it, and seeing advice for how to deal with it well might help you with a character who "can kind of hide it."
And I know everyone's gunshy about tumblr but like.... seriously, search tumblr for posts tagged with that mental illness. Take stuff with a grain of salt, because there's no way to know which posts have experience or expertise behind them, but when you come across someone talking about their actual experience to a community of people that want to support them, that's tremendously valuable insight.
The other thing you might find there is that different mental illnesses will kind of have different "stigmas" associated, so there's no across the board "here's the right way to portray mental illness in general." For example, OCD is often portrayed as a silly joke, so a lot of people with OCD really want others to see that it is an actual hardship and not a funny personality quirk. Whereas "psychotic" illnesses like schizophrenia tend to be portrayed as tragic or frightening, so a lot of people who actually experience and deal with these mental illnesses really want to see psychotic characters who aren't tragically afflicted.
And I mean, you can't please everyone and that's not your job, but seeing people say "here's the thing everyone leaves out when they make characters with my experiences" is super useful insight either way.