you can check out Bad Guys.One of the main characters supposedly keeps committing murders but he doesn't remember doing it. there will be a sequel for it this year. my one and only oppa park hae jin plays one of the characters.
I am currently watching Strong Woman Do Bong Soon and trust me when I say there is a psychopath in this one. This drama is still airing, but I can't really recall any others that haven't been mentioned already above. ^_^
How about Cheese in the Trap? Mental illnesses are not specifically adressed but based on how those characters behaved i am pretty sure that at least 3 characters do suffer with some sort of mental illness. Seriously, it has many messed up characters
Definitely agree with White Christmas as recommendation. It not only fits perfectly what I think you want but is also just in general a great show imo. I see that you have Empress Ki on hold, you could say that this one also has a mentally unstable character or two ^^ maybe give it another chance. The K2 was somewhat of a mess but a lot of people got mesmerized by one of the female characters that was definitely psychotic. Ms. Perfect that is airing right now actually also has a psychotic character, the show itself is rather crazy though - it's walking the fine line of Makjang - I personally can't complain though - it's thoroughly interesting. All in all though on the lighter side and not serial killer dark. I guess you could maybe say that Bridal Mask has psychotic characters. lol I think you would at least find it interesting to analyze the main character's psyche. Gap Dong definitely has a psychopathic character though I personally didn't watch it until the end but that doesn't mean it couldn't be for you. If I remember correctly, I just felt like watching something lighter instead so that's mostly the reason I dropped it.