Yes, you can still become a vampire. Health potions (potions of Healing or Health) can't cure diseases. (The 'Cure Disease' potion can, however.)
You should still have Sanguinare Vampiris even if you used a health potion, and will eventually progress to being a Vampire, provided that you don't cure your Sanguinare Vampiris disease.
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.