Before you become a vampire, it is recommended that you become a member of the Mages Guild, access the Arcane University, and either become a member of the Dark Brotherhood or started the Order of Virtuous Blood quest.
Dark Brotherhood Story
Become a member of the Dark Brotherhood by killing an innocent person, going to sleep in an inn, traveling to the Inn of Ill Omen, and killing Rufio. Then, travel to the Sanctuary and speak with Vicente and finish all of his contracts. He will then ask if you wish to become a vampire. Say "Yes", then sleep on the slab. When you wake up, sleep again for twenty four hours three times to be a full-fledged vampire.
Order Of Virtuous Blood Story
It is not required to start the quest, but if you do, the vampires will be easier to kill. If you have not, travel along the river just to the east of Imperial City until you find a cave. Go in and you should find plenty of vampires. Attack one and let it hit you a couple times. Kill it, then go back to Imperial City. Get a room in an inn and sleep for twenty four hours three times. You are now a full-fledged vampire.
To become a vampire without doing quests, when you sleep outside frequently, eventually you will wake up and a message will state that you have had a strange dream. After this, you will have very strange dreams and will have all the attributes of a vampire. Be careful of being outside during the day.