Disclaimer:
Following this path in the Dark Brotherhood questline will result in the player to lose out on a lot of in-game gold and the option to do some pretty fun quests. That being said, the alternate path adds a nice twist into role-playing characters.

Start the Dark Brotherhood questline as normal by visiting the Aretino Residence in Windhelm and completing the quest "Innocence Lost" given to you by Aventus Aretino.

Wait 24 hours. You'll be given a letter saying "We Know". Then sleep.

After waking in the Abandoned Shack, you will see Astrid atop a bookshelf. When speaking to her always refuses to kill the three other prisoners in the shack. Eventually Astrid will say, "You don't leave here 'till someone dies." or something like that.

At this point you will attack and kill Astrid. Upon her death she will say "Well done". Meaning that the correct person to kill was her.

Take the key to the shack and leave. Note: It would be nice to free the other prisoners before leaving, but that is up to you.

Report Astrid's death to a guard this will start the alternate Dark Brotherhood questline. Follow the quest objectives and there you go.

Have fun. I hope this helps.