Hrm... I have to say that I often base my characters on myself, to a point, which assures me their personality is somewhat believable. Some quirks, flaws, mannerisms, etc, they can really flesh out a character, and it's as simple as looking at the people around you our yourself.

As for backgrounds... I always like having a detailed past. It forces me to be sure of what I am saying, do the necessary research before I say I'm so-and-so from Genericville. It is important, however, to give your character space to grow as you play it. Sometimes you find that an idea you had in mind doesn't really fit in, and you begin to tweak here and there.

Basically, I'm saying that I like a solid base to stand from, but I also make sure I have a ceiling that's tall enough for me to develop the character I'm playing with freedom.

My pc's come out of my subconscious with no rhyme or reason. Sometimes I just want to get as far away as possible from what I did last. Sometime I wonder what if someone...? Then I figure I might as well find out. It's totally random.

These days when I play, which isn't terribly often, I often pull characters verbatim out of "The Silmarillion". Of course this is on less rp-oriented muds, if I wanted to play house with some 40 year old truckdriver living in his parents' ****-soaked basement, I'd just go do that and save on the electric bill of MUSHing. But back to silmarillion, most the characters in there have their own plethora of names, so one can pick one of the less well-known of these, and that way it's almost like a "secret" that you're ripping them off (for example, play as Thingol but name yourself Elwe). Most the character types worth playing (minus minor variations and eccentricities) are covered in silm, and it saves a lot of effort coming up with your own traits (holy crap, some people put so much effort into this "hobby" that it's obvious they have absolutely no life and that they have never actually worked a day during their miserable worthless existence!)

Inspiration is always fun to get. Though my problem tends to be it wakes me up one day... then I have to wait a week to put it through the crucible of 'is this really a good idea?' and 'will it be playable?' That takes a lot of ideas out right there, or at least puts them into NPCdom.

Anyway, my best characters are the ones I tended to think about, usually with a basic framework bio. (I'm talking around 500-750 words or so, nothing elaborate, but I do like to think about a character before playing)

Though along with that, I don't like wasting time investing into bios and ideas, which is why I do my screening. So most of my characters tend to work quite well.

My best characters though? I have no idea how I find them, in the end. I've found them from random inspiration, filling holes of not having enough characters... though one of my current favorites is an NPC that took on a life of his own.

I'm also generally a fan of a bio or something so I know where my character is going. Though I have done some in depth research for a couple ideas.