Giving your Character Soul
The best way to do a character background, methinks, is to just jot down the major events, so you get what I call a 'skeleton'. This gives your character form, but doesn't restrict things you might want to add later.
A big point, I think, is which 'age' your character belongs in. Is he of modern origin or was he from a long time past? On a personal note I think that if you haven't played before it's better to play a 'modern' vampire, which means you don't need to make up lots of history on what you have done relating to other vampires. This shouldn't restrict you in any way at all, because an awful lot did happen this century. We had two world wars for a start, and several dramatic cultural changes due to massive leaps in communication.
The second big thing in vampire, I think, is your sire. Now in most cases, Embracing a mortal is not something done lightly, especially amongst Toreadors. Why did you think your sire embraced you? Is it for company? To continue his work? Just a vague idea is good, because the more blanks you leave about your sire, the more flexibilty you have later when you get ideas. Similarly, have a think about which era your sire came from, because that will cloud his way of thinking too. And while you're at it, is he still around?
Once you've got these two points, you will have a time reference to work on. Just look up what major events happened over those period of time (and major events are relative to the observer) and see what your character did over that period. e.g. if he was born about 1930s, did he fight in WW2? What does he think of it? How was he affected by the Beatles? etc, etc, etc,.