Where is the Dark Brotherhood?

Solution 1:

Head to Rorikstead and ask the inn keeper about rumors. He'll inform you about a kid. (The kid is Aventus Aretino in Windhelm, if you want to skip the rumor gathering) This will start the dark brotherhood questline.

Solution 2:

Another option to start the Dark Brotherhood questline can be found in Riften. Head to the Honorhall Orphanage and listen to the horrible, disgusting woman who runs it as she belittles and verbally abuses the children inside. Upon viewing this small conversation, you can then speak to one of the children in the orphanage. They will tell you about Aventus Aretino, and ask you to find him. This also begins the Dark Brotherhood quest.