I love 'psychologically challenged' characters, whether they are heroines, protagonists, or antagonists.  That said, I don't mean those cookie-cutter 'villains' and 'serial killer' types when I say this.  I mean characters who are twisted up inside and as a result have a completely different outlook on life or appear inconsistent on the surface as a result.  Villain and serial killer types, who just cause harm because they want to, just aren't that interesting.  Here is a shortlist of good VNs that also have a few psychotic characters that are interesting to me.

Ojou-sama wa Gokigen Naname (Yuuzenji Tsurumi)- Antagonist of the story, she is passionately and with utter seriousness in love with money.  She will honestly and totally love anyone who gives her money, but that love disappears the moment that they have nothing left to give her.

Kikan Bakumatsu Ibun Last Cavalier (Okada Iori)- Iori is the partner heroine of Ryouma.  She is the very picture of someone twisted by her job, as she is an assassin for the anti-Shogunate forces who is more than a bit yandere.  Her twisted need to be controlled and have sex with the one who tells her to kill is a primary personality trait for her.

Devils Devel Concept (all heroines and Sora)- Sora is a young man with a domineering and tyrannical personality and an obsession with transactional interactions with others in the form of 'contracts' (not to mention a psyche that reacts to violence, pain, and inflicting pain with joy).  All the heroines are a bit broken in various ways, though each has a different broken point.  There are also a lot of other broken characters, lol.

Draculius (Jun and Belche)- Jun is normally an easygoing and kind-hearted young man, but he is insanely protective of 'family', to the point of extreme ruthless cruelty if he feels it necessary.  Belche is an ancient Irish vampire who bases her entire worldview on her service to a master and just how much she can give to him, while also being a coldly ruthless monster who would willingly kill entire towns if her master asked it of her.

Grisaia trilogy (seriously, does anyone need me to point out all the broken characters here?)

Comyu (Kagome)- ... anyone who has played this game knows Kagome is crazy, so I don't know any reason why I should continue.

Silverio Vendetta (Zephyr and Valzeride)- Zephyr is a cowardly deserter who is normally happy to drink himself into a stupor and engage the services of easy women, but when cornered he becomes a sadistically brutal monster who will happily torment and slaughter those who oppose him using any tool that comes to hand.  Valzeride is on the surface the ultimate, perfect patriotic hero.  However, his hard-working nature goes beyond the norm, as he is quite literally incapable of not pushing his own limits and enduring the most hellish of experiences solely to be the 'Enemy of Evil' as he terms it.  He fundamentally loves destroying evil as the root part of his nature, and he naturally seeks to turn the ruined lives he leaves in his wake into the base for his nation's prosperity.

Silverio Trinity (Gilbert and Dainsleif)- Gilbert is a hero-obsessed strategist who always has a scenario ready to get the best result out of any situation.  Dainsleif is also obsessed with Valzeride and those like him, desiring nothing more than to be the one to oppose and destroy heroes as a self-styled evil dragon.

Dies Irae (Reinhard Heydrich and friends)- That there is something wrong with the psychology of every single person on Reinhardt Heydrich's side, as well as most of Ren's side is a self-evident reality.

Tokyo Babel (Setsuna) - Seriously, talk about overweening arrogance hidden in a huge lump of self-pity and self-hatred, lol.  

Yami to Hikari no Sanctuary (Souji)- With Souji it is mostly a result of his upbringing, but he seems so normal... even as he kills.

Yurikago yori Tenshi Made (everyone)- No normies in this game, nothing but psychos.

Izuna Zanshinken (lots of people)- Lots of crazies and oddballs in this game.

Vermillion Bind of Blood- Toshirou, Ariya and Tri-finger.  Figure the rest out yourself, lol.

Hapymaher (Maia)- Say what you want about Maia, but she is probably one of the most internally conflicted characters I've seen.  The reasons make perfect sense, if you know what to look for.  Tohru is a bit crazy too (anyone who has to fight that much with his own desire to dream forever is a bit brain-screwed).

Butterfly Seeker- Another game with a lot of messed up people.

Psychologic Love Comedy- Craziness everywhere

All the Smee Seasons games (Harumade, Natsukumo, etc)- craziness is this series' selling point.