It seems that I'd like a character more if I were to be like "hey! I want to know someone like that!"
Though it does seem like I have a huge weakness towards the "troubled past, occasionaly brooding, and eventually turns out to be an extremely loyal friend" type of character. It usually amuses me or interests me on how the main character (typically) reaches out to them. Most prominent examples off the top my head is probably Fate Testarossa from Magical Girl Lyrical Nanoha and Mai Kawasumi from Kanon. May or may not contain the entire main cast of Higurashi.
You get zero points for figuring who I'd like in Umineko.
Someone related, is just the simply detatched and mysterious type that proves to have a lot more to them, such as Yuki Nagato.
Another favorite character type of mine are characters that are extremely confident in their abilities and always manage to do things with some kind of class, whether it be general behavior or simply pwning whoever asks for it. Some of these people include Tomoyo from Clannad, Kurama from Yu Yu Hakusho, and Shigeru Motherfucking Akagi from Akagi. The battle is won before it began. Well, I suppose Kenshiro from Fist of the North Star belongs here too, huh?
I also like characters that have very strong ideals and goals and the determination to uphold these values; the opposite would be a self righteous hypocrite. Celty from Durarara!!or Yuri from Angel Beats! are very admirable characters to me because they are so level headed about it as well. Nanoha, is a great example, simply because she is also so practical and balances it out very well.
And of course, Tomoyo can fit here too, which is probably why I have an avatar of her.
And since I'm a huge fan of sarcasm, I like those characters much-- Kyon is the man for that, of course.
For villains, I like those that take pride in their role- trolls. Izaya from Durarara!! or Beatrice from Umineko amuse me greatly.
In general, the ones I like the most will overlap a lot in these categories. Characters I hate are usually hypocrites, preachy, annoying, inconsistent, or display a general lack of empathy of others (especially if they are supposed to be a protagonist).
I have no problems with a character being weak or doing stupid shit , but I do when they are supposed to know better.