I've already become a werewolf in Skyrim, but how do you become a vampire? Is there anything special I need to do, or things I shouldn't do?
You cannot become a vampire while you are a werewolf because werewolves are immune to all diseases, so you cannot contract Sanguinare Vampiris
If you cure your werewolf issue, then you can contract the disease simply by allowing a vampire to do damage to you until you receive it.
Afterwards, idling for 72 hours will turn you into a full fledged vampire.
You can be a vampire while you are a werewolf only with the Dawnguard expansion. Serana or her father can turn you either way.