This quest is obtained later in the game, after completing several side quests for the Vampires. When you speak to Garan Marethi in Castle Volkihar or to one of the other vampires, and ask what you can do to help, they will refer you to Fura Bloodmouth for a side quest.

Fura tells you that the Dawnguard will want the Bow, and they will fight us, hard, to get it. We should strike them first before they can muster a larger force. They've holed up in an abandoned fort in the Reach. Kill their leaders. That should put a stop to a full out war.

This beings the quest:

Destroying the Dawnguard
Kill the Dawnguard Leaders

The fort is in Dayspring Canyon, southeast of Riften.

Go to Fort Dawnguard and enter it. Kill all the Dawnguard leaders including:
Florentius Baenius
Isran (usually Isran cannot be killed since he is a story NPC, but in this quest he can be killed)
Sorine Jurard

If possible, kill them stealthily and quickly. If they manage to group, they will become extremely hard to kill.

Note: You can cast Pacify on NPCs here and get some cool stuff which is usually Dawnguard exclusive. For example you can buy the spells Stendarr's Aura and Vampire's Bane from Florentius Baenius, Sun Fire from Sorine Jurard, and an Armored Troll from Gunmar.

Return to Fura Bloodmouth in Castle Volkihar after you have killed all of the Dawnguard Leaders in Fort Dawnguard. Tell her you have killed the Dawnguard Leader.

This completes the quest, Destroying the Dawnguard