I have three different characters that have become vampires at one point or another, all for different reasons and they all handle it differently.
There's Andreil Riztezze, a Breton thief in the Thieves Guild who got inflicted with the disease while trying to steal an artifact from a cave. She was expelled from the guild, which was her only home. After that, she retreated into the wilderness and never fed on anyone. Insanity started to close in, as it often does to vampires who do not feed, and she knew she had to cure herself of the disease. She tracked down the Count of Skingrad in his bedroom and got the information of where to find a cure for vampirism and succeeded in curing herself. Her roleplaying during that time mostly consisted of hunting down animals, such as deer (with the idea being that she'd feed on the blood of the animal, although much much less satisfying than that of one of the 10 races.) It also involved dodging Imperial patrols on the roads and finding a nice hideout.
Then there's Cyrian, an Altmer scholar of all things Dwemeris in Vvardenfell. Cyrian was a tad bit crazy to begin with, and, as age started to take its toll on him, and after getting badly injured in the Dwemer ruins under Mournhold, Cyrian came up with a solution as to how he could live forever. He intentionally tracked down a vampire hideout where he asked one of the vampires there to turn him into one of them. He got lucky, as typically most vampires would have attacked him on sight, but he asked just the right vampire, who agreed to his request. Cyrian isn't really one to spend much time in towns and cities, so he rarely had a chance to feed. Because of this, and because of his already fleeting mentality, insanity quickly set in. He's developed a full-on obsession with Dwemer ruins and studying them, and will attack anyone he sees on sight. Roleplaying as him pretty much involves being crazy, and it's pretty fun!
Finally, there's the vampire Imperial woman who has long forgotten her birth name, but adapted the nickname "Misery." She was turned into a vampire back in the Second Era. She embraced the gift and developed a thirst for power. However, Misery knows the dangers of displaying her powers in public, so, instead, she frequently manipulates other people to do her bidding for her. Misery is often mentioned in the history of Tamriel, though never by name, and a connection between her places in history are never linked. She was the Nightblade that slit the throat of Tiber Septim and burned the Imperial Palace. She was the master behind the Camoran Usurper's invasion. She also was part of the Imperial Council during the rule of Jagar Tharn. And that vampire I mentioned earlier that agreed to turn Cyrian into one, that was Misery too! Roleplaying as Misery is a lot different than anyone else for me. She's very powerful but almost never displays the extent of her power in public. She feeds very very frequently, and, being of the Cyrodilic strain of vampires, this allows her to mask her identity as a vampire from the general public. Most of the time she wears middle-class clothing in order to blend in with the rest of society.
Hopefully that gave you a few ideas of different ways that you can take being a vampire. As has been said above, a lot of it depends on who your character was
before becoming a vampire, as that's what dictating how all of my characters turned out.