Destroy the Dark Brotherhood begins after meeting Astrid, the leader of the Dark Brotherhood, in Skyrim, upon failure of With Friends Like These, in which you are given a choice of three captives to kill. This quest triggers if you instead fight and kill Astrid.
After dispatching Astrid, you are told to report to any guard with the information that you have killed a Dark Brotherhood assassin. He sends you to Commander Maro in Dragon Bridge who then gives you the subquest Kill Everyone in the Sanctuary.
You get a fair amount of expensive gear from the assassins. When you report back, Commander Maro gives you a reward of 3000 Septims.
Finishing this quest removes all the quests for the Dark Brotherhood faction for the remainder of the playthrough.