Now, to continue with this questline, go head back home (or a secure bed somewhere) and sleep for awhile (I slept for 24 hours, just to be sure). You SHOULD awaken in unfamiliar territory and someone named Astrid will ask you if you slept well. She’ll tell you that by all rights, Grelod was a Dark Brotherhood contract… a contract you stole, which means that you owe her a kill. She has asked you to kill someone in the shack that she took you to in order to earn your freedom and repay your debt.
You’ll find three hostages behind you. Now, try as you might, there is NOTHING here that’s going to tell you WHO has the contract out on them. You can talk to each of them to hear who they are (and get some speech skill opportunities) but in the end you must kill one of them.
Personally, I killed all three, as they all seemed like they could be guilty (and Astrid will also make a different comment depending on how many of the three you actually kill - which is a nice touch).
Note: Instead of killing one of the captives here you COULD turn around and kill Astrid. Doing this will FORFEIT you being able to do ANY of the Dark Brotherhood questline though! However, it does start a seperate quest called "Destroy the Dark Brotherhood" . If you want my opinion; the Dark Brotherhood questline is much more interesting and profitable…
Go talk to Astrid now who will congratulate you. She’ll also say that it’s not a matter of WHO has the contract out on them: she told you to kill one and you did. She’ll give you the key to the shack and will then tell you to seek out her family in the Pine Forest (which is to the south, by Falkreath). The Dark Brotherhood Sanctuary is there. She’ll also give you the password to the black door: "Silence, my Brother".
Head over to Falkreath and head west to The Dark Brotherhood Sanctuary (which is pretty easy to find and not very well hidden if you ask me). Give the door the password: "Silence, my Brother" , and head inside. You’ll find Astrid here so talk to her to officially join The Dark Brotherhood. She’ll give you a welcoming present (a set of their clothes) and this quest will end. The next Dark Brotherhood quest, " Sanctuary ", will now be triggered.