I'm not sure that you can stop the assassins by destroying the brotherhood, but here's the basic of doing it anyways.
Go to Windhelm and search around for a Dark Elf talking to a kid, if you listen to them you get a quest marker to talk to some person called Aventus Aretino (you don't need to hear them, you can just go to the house; it's the one right next to where the two are talking, you need to pick the lock.)
Inside talk to the kid and start the Innocence Lost quest, where you need to kill the person specified in the quest. You don't get a bounty even if you kill the person in public. Afterwards you will receive a letter from a courier (after a FEW days, either do some other stuff or wait/fast travel).
Afterwards, sleep in any bed for any amount of time, which will start the first quest for the Dark Brotherhood, in which you can decide rather than to follow the orders of the Brotherhood leader, to just kill her instead. Be forewarned, she has one of the strongest daggers in the game and some very good armor, so it will be a bit of a fight.
Afterwards you will fail the first Brotherhood quest and get a new quest to destroy the Brotherhood, find a guard (any city will do), tell them you've killed the leader of the Brotherhood, and you'll get a quest marker to talk to someone in the town of Dragon Bridge, who will then give you the location (and password) for the Brotherhood hideout, where you can then go and kill and get a few thousand gold (plus the loot, including some good light armor and one of the better daggers in the game).