It is possible to become a vampire in the world of Skyrim. I don't recommend this as a first play through as it can make the game a lot more challenging, but if a challenge is what you are looking for, here's how you can become a vampire.
Step 1)Contract the Sanguinare Vampiris disease.
Morvarth’s Lair is probably one of the easiest places to go to catch this disease. To catch it you'll have to be hit by a vampire, and it can take a few hits. If you are already a werewolf you can't contract the disease at all as you are immune, and wood elves and argonians are both going to have a harder time contracting the disease due to racial resistances. Just let the low leveled vampires wail on you here until you do contract it, and keep healing so they don't actually kill you while they do it. As long as you aren't a werewolf it'll happen eventually.
Step 2)Don't cure the disease.
The disease can be cured by praying at a shrine, or by drinking a cure diseases potion. Just avoid doing either of these things for 3 in game days and the disease will progress into full blown vampirism.
Step 3)Enjoy your new vampiric nature!
You'll need to feed regularly to avoid letting your vampirism progress. If it gets to stage 4 you will be obviously a vampire to anyone who sees you and they will attack on site. You'll also be stronger against frost and gain more abilities, but be weaker to fire and sunlight the longer you go without feeding.
Remember, sneaking into people's houses at night is the best option to feed, so be sure to invest in your sneaking if you plan on turning to a vampire and want to avoid getting attacked on sight.
Step 1)Contract the Sanguinare Vampiris disease.
Morvarth’s Lair is probably one of the easiest places to go to catch this disease. To catch it you'll have to be hit by a vampire, and it can take a few hits. If you are already a werewolf you can't contract the disease at all as you are immune, and wood elves and argonians are both going to have a harder time contracting the disease due to racial resistances. Just let the low leveled vampires wail on you here until you do contract it, and keep healing so they don't actually kill you while they do it. As long as you aren't a werewolf it'll happen eventually.
Step 2)Don't cure the disease.
The disease can be cured by praying at a shrine, or by drinking a cure diseases potion. Just avoid doing either of these things for 3 in game days and the disease will progress into full blown vampirism.
Step 3)Enjoy your new vampiric nature!
You'll need to feed regularly to avoid letting your vampirism progress. If it gets to stage 4 you will be obviously a vampire to anyone who sees you and they will attack on site. You'll also be stronger against frost and gain more abilities, but be weaker to fire and sunlight the longer you go without feeding.
Remember, sneaking into people's houses at night is the best option to feed, so be sure to invest in your sneaking if you plan on turning to a vampire and want to avoid getting attacked on sight.