Well, what ideas do you have for the background of your character?
Did they choose to become a vampire? Was it an accident and they couldn't reach help in time?
Why is your character an assassin? Money? Religious beliefs? Insanity, perhaps?
What was their childhood like? Growing up in the streets all alone? Trained in the ways of an assassin by someone or some organization?
I've actually thought about this before.
Well he's a Dunmer named Andural Relamus, but his family was always somewhat poor, so when they came to Cyrodiil they were farmers. His father trained him to be an exceptional Marksman, Swordsman, and also to Sneak very well while stalking prey for their dinner. Sadly, his mother soon died by a Bandit ambush, and his father by sickness. He was an only child so now it was up to him to keep the farm alive.
One early morning while hunting, he came across a Bandit camp. thinking these could be the same ones that killed his mother, he slowly walked closer and then noticed someone crouching over one sleeping bandit. Noticing it was a Vampire let out an uncontrolably, but quite, gasp. The Vampire noticed and lunged at them. Being too slow, he was pinned down where the Vampire then proceeded to bite into his neck. He awoke days later by a strange dream and noticed he was desperately thirsty. But not for any Wine or Mead, but for Blood. He then ran home as fast as he could and began hiding in his home.
One day a nieghbor farmer's watermelons were stolen and placed in Andural's Yard. But he had not taken them because he was still locking himself inside. This farmer then called the Imperial Gaurds and blamed Andural for the crime. Once they came they asked him to pay a simple fine. Andural had been short on Gold for a while now, due to lack of working. So he tried to make the Guards believe him that he had nothing to do with it. Well they wouldn't listen and seemed to greatly dislike him from his new appearance. He ended up losing his temper and pulled the nearest Guard's Sword and stabbed him with it. They didn't like this. So Andural was then sent to the Imperial Prison.
Once he finally escaped, he went to quench this thirst. He went back home and in the night drank the nieghboring farmer's, who accused him of the theft, blood and then proceeded to stab him multiple time in the heart, while watching the blood drop to his feet. He went to his old home and slept, only to be awoken by Lucien Lachance who had a very generous offer. He was then captivated by the arts of the Assassin and was greatly pleased by his acceptance even though of his monstrosity.
And that's about as far as I've gone.
I'm thinking about something where he has to track and kill the Vampire that gave him this curse. But I can't think of an NPC that would be good for it.