If you don't have Dawnguard, the only way to become a vampire is to fight a vampire. Any time a vampire uses their Vampiric Drain spell on you there's a 10% chance you'll contract Sanguinis Vampiris. As it says on UESP (the Unofficial Elder Scrolls Pages wiki; just about the best source of Skyrim info out there): "Initially, this is a normal disease that reduces your total health by 25 points and can be easily cured with a potion of Cure Disease, a Vigilant of Stendarr, or by visiting a shrine. At sunrise of the first day after contracting Sanguinare Vampiris, a message will appear saying 'You feel strangely weakened by the sunrise'. At sunset of that day, 'You feel a strange thirst come over you'. If the disease is not cured within three days, it will progress to full vampirism, and the effects can no longer be reversed with a normal cure."
If you have Dawnguard, you can be made into a Vampire Lord by Lord Harkon or Serana just by progressing through the DLC story.