Skyrim:Rising at Dawn

Quick Walkthrough
# Talk to any innkeeper.
# Go to Morthal and talk to Falion.
# Bring him one filled black soul gem.
# Meet Falion at the summoning circle outside of Morthal.
# Watch Falion perform the ritual.

Detailed Walkthrough
This quest can only be started once you have become a vampire, either through the incubation of Sanguinare Vampiris or through infection by a member of the Volkihar clan. It is repeatable and there is no limit to the amount of times you can perform the quest.

Talk to an Innkeeper
As a vampire, ask any innkeeper if they have heard any rumors. You may need to ask several times before you get the rumor you want. This can be quite time consuming. They will eventually point you toward Falion, a mage who studies draugr, vampires, and other undead in Morthal. Sometimes he will not discuss vampirism with you until you've been told to go see him.

Travel to Morthal to Talk with Falion
Travel to Morthal, a town northwest of Whiterun. Falion lives in the easternmost house in town, with his ward. When spoken to he will reveal that he studies the undead (under some amount of disapproval from the town) and that he recognizes you as a vampire. He will offer to remove your vampirism if provided with one filled black soul gem.

Finding a Black Soul Gem
Falion and Enthir are the only merchants who are guaranteed to stock black soul gems. The gems also appear randomly, dependent upon your level, on warlocks and in geode veins.  Note that The Black Star will not work. As Falion requires a filled black soul gem, you will then need to fill the soul gem by soul trapping someone. The nearest location with enemies is Fort Snowhawk to the west of Morthal, or Movarth's Lair to the northeast (if the fort has been taken by the Stormcloaks or Imperial Legion).

Meet Falion at Dawn
At 3am, Falion will start making his way to the circle and remains there until 7am. Meet him at the summoning circle to the north of town at dawn. He will perform the ritual, during which the screen will fade to black before returning, and you will no longer be a vampire. If you miss meeting him, reload a previous save and retry.

Notes
If you are a stage four vampire, Falion will not perform the ritual, and may attack you. You may use Calm to make him non-hostile.
This isn't an issue if the Dawnguard add-on is installed.
If you have the Dawnguard add-on and join the Volkihar vampires, you can still cure yourself by completing this quest. You will lose all your vampire abilities and the Volkihar will no longer talk to you but will not become hostile.
Recruited members of the Dawnguard will direct you to Falion if you become infected throughout the course of the Dawnguard questline.

Bugs
 
In order to complete the objective, you must force him into the circle, either by luring him into the circle by attacking and calming him or by nudging him little by little into the circle (fist punching and sheathing fists will encourage Falion to attack instead of running away). Once he is in the circle, speaking with Falion will trigger the next stage of the quest.
  This can by fixed by clicking on Falion and typing   in the console.  You must be in the circle since he will be teleported next to you.
 
  This can by fixed by typing   in the console.  This forces the quest to be completed and vampirism will be cured.

Quest Stages