I always play as a female character. I don't know why but picking male characters never enters my mind. It doesn't matter what the game is, if there's a choice of gender then I automatically go for the female option(s). If I have a say in their look, I go for a gothic or alternative style; I never pick the blonde-haired, blue-eyed fantasy hero with the big shiny sword. Typically my custom-created characters have dark or red hair and feature lots of purple. I try to avoid scantily-clad characters, though in fighting games this seems impossible.
Following that, I tend to play in a stereotypically "video game female" way, e.g. using bows rather than swords, speed rather than strength; in the case of games like Fallout I try to avoid combat and talk my way through missions if possible, etc. Again, I don't know why, but it has just never occured to me to play any game in the typical big brute style. I tend to dabble in magic/psychic/etc abilites but usually in a support role. So for example in Guild Wars, where you can pick a primary profession and a secondary, my main character is a primary Ranger (bow) with a Mesmer (support magic) secondary. In Mass Effect my Commander Shepard is an Adept, mostly using sub machine guns and sniper rifles and occasionally using a psychic power if things get really hairy.
As for personality and morality systems, it really depends on the game. In Fable I play once as all-good and once as all-evil; in Mass Effect 2 I pick options depending on the situation and end up overall being in the middle. The only exception is if there is an option to be a character who is out of her mind, in which case I pick that and just roll with it, regardless of the implications that may have on gameplay. This is rarely an option in moral choice or character creation modes since that's usually all boiled down to good-vs-evil, but sometimes premade characters will turn up like Tira in Soul Calibur who I absolutely love. I suppose I like the notion that a character isn't behaving in a particular way because they want to be good or bad, but more 'just because'. I like shades of grey and being able to do what I want in the moment rather than being limited to one side or the other, and unhinged characters tend to fill this spot.
In short, I am Harley Quinn; my natural inclination and place is to be a supporting female character who is not wholly evil and chaotic but certainly random as can be.
edit: and in case you can't tell, I always play as a character who writes way too much.