Destroy the Dark Brotherhood!

Walkthrough
After killing Grelod the Kind and deciding to have a good night's sleep at the Sleeping Giant Inn, the Dragonborn wakes up inside the Abandoned Shack, with Astrid, who turns out to be the leader of the Dark Brotherhood.

She explains that Grelod was by rights a Dark Brotherhood contract and that the Dragonborn has infringed on their territory. She then explains that the Dragonborn must fulfil a contract for the Brotherhood as recompense and indicates three captives who are blindfolded and bound, telling the Dragonborn that they must decide which captive is destined to be assassinated and kill them, but she will not tell you which one.

Instead of doing this, the Dragonborn can choose to kill Astrid.

Report Astrid's death to a guard
Once Astrid is dead, grab the Blade of Woe and the Abandoned Shack Key from her body. The three prisoners can now be freed.

The Dragonborn then needs to locate a guard and report her death. The easiest way to find a guard is to go to the nearest city. The guard will then instruct the Dragonborn to speak with Commander Maro in Dragon Bridge.

Speak with Commander Maro
Commander Maro is at the Penitus Oculatus Outpost in Dragon Bridge. He will provide the Dragonborn with the location of and the necessary pass phrase ("Silence, my brother.") to enter the Dark Brotherhood Sanctuary.

The Dragonborn then heads off to the location and uses the pass phrase to gain entry.

Kill everyone in the Sanctuary!
Once inside, the Dragonborn must kill all the residents of the Sanctuary.

Report back to Commander Maro
Once all the members of the Brotherhood have been killed, the Dragonborn returns to Commander Maro to receive a reward.

Zerstöre die Dunkle Bruderschaft!