The easiest is to find Aventus Aretino in Windhelm. Break into his house and you'll see the boy performing the Black Sacrement. He basically wants someone assassinated and called upon the Dark Brotherhood to do the job. He sees you and mistakes you for being a Dark Brotherhood assassin. You go to Riften to kill the lady who runs the orphanage because all of the kids hate her. After you kill her, a courier will randomly appear after about some game hours and give you a letter with a bloody hand print and a statement saying: "We know". After you receive this letter, go to sleep for however many hours. You'll wake up in an abandoned shack with Astrid (leader) and three hostages. She basically tells you that, since you stole the assassination which was meant for the Dark Brotherhood, you have to repay the stolen kill by killing one of the three on the spot. After you killed one, two or maybe even all of them, Astrid will invite you to join the Dark Brotherhood. But also note, this is the only point in the game where you can destroy the Dark Brotherhood. You can attack Astrid here without having her become invincible.