To even get this quest you have to first be a VAMPIRE! That's right, this quest right here is, ultimately, the quest you want if you are a vampire and you want to cure yourself of the condition.
As a vampire you want to head to an Inn (doesn't matter where) and ask about rumors until they tell you about how Falion in Morthal is studying vampires, at which point the quest will automatically start (I would imagine you could go straight to Morthal and talk to Falion as well, but I didn't test it).
Follow the marker to Falion's house in Morthal, where he will get very defensive about his research. Ask him about his vampire research and he'll tell you about it and strongly hint that he knows what you are. Next ask him for a cure and he'll advise you that you'll need to bring him a full BLACK SOUL GEM .
Thankfully, Falion just HAPPENS to have a black soul gem that you can buy off of him (mine cost me 149 gold, which is CHEAP). I STRONGLY recommend buying the gem off of him in comparison to finding one yourself. Our next step in this process is to FILL the black soul gem with a soul.
Black soul gems are gems filled with human souls , by the way (regular gems are non-human souls). What we need to do is to find some bandits or other humans and soul trap them with either the soul trap spell ( which Falion ALSO sells, by the way ) or a weapon designed to capture souls (which you'll run into a LOT of as you play the game).
No matter HOW you get it, once you get it talk to Falion once more and let him know you have one. He will tell you to meet him at dawn. The place you are going to meet him at is nearby the city, just to the north, on top of a ritual circle, and you want to meet him there around 4-5 AM (dawn).
You ALSO want to be a stage 1 vampire when you meet him, which may very well mean going on a feeding run before the ritual. Falion will outright NOT DO THE RITUAL if you aren't a stage 1 vampire (I was stage 2 and he didn't do it for me!).
Once you have that taken care of Falion will come out. Talk to him and he'll start the ritual, which merely involves you blacking out for a bit but at the end you will no longer be a vampire.
Note: As ungrateful as it sounds, after the ritual you can talk to Falion near this circle and question what he's doing out here (which YOU of all people should know!). He will pay you 200 gold to forget that you saw him out here...