To join Dark Brotherhood, make your way to the Aretino Residence in Windhelm. There, you will see a boy named Aventus Aretino who is trying to do a ritual. He will give you a quest called "Innocence Lost" which will lead you to kill Grelond the Kind.
After doing this and finishing the quest, thus starting the Dark Brotherhood questline, you will either have to wait 24 hours or so for a courier, which will hand you a letter with a black hand print and the words "we know" on it, or sleep in an owned bed to get kidnapped.
Here is the crucial point, when you are kidnapped by Astrid, this gives you the quest "With Friends Like These..." where she wants you to kill a captive.
Make sure you agree to do this and kill the captive or all three if you want and do not, I repeat, do NOT attack Astrid. Attacking her will give you the quest "Destroy the Dark Brotherhood" and thus voiding the achievements related to this quest line.
Once you kill the captives, Astrid will tell you to go to the black door that is in southern pine forest, there you will be asked by the door about the sound of music. Pick the first option "silence, my brother" and enter. Go talk to Astrid, the quest will finish and the achievement should pop there.
Hope this helps.