I was trying to become a vampire by letting Alva hit me with her vampiric drain spell but people were also shooting me with arrows and I was about to die, so I drank some minor health potions to survive. Can I still become a vampire or do I have to try and catch the disease again?
Whenever a vampire casts Vampiric Drain on you, there is a 10% chance for you to be infected with Sanguinare Vampiris.
However, rather than letting them hit you in melee, you should keep your distance and only allow them to cast their draining spell as this is the only attack that can impart the disease. To speed up contraction, repeatedly lose and gain sight of the target vampire so the spell is cast multiple times. The check for giving you the disease is passed only when the draining spell hits you, not over sustained action.
If you have already contracted Vampirism (gone beyond the third day of infection of Sanguinare Vampiris), you'll need to talk to Falion and give him a filled Black Soul Gem to cure yourself of Vampirism.