You can do it by completing the Arena questline and needed to do the Origin Of The Gray Prince side quest and you need to do the side quest before you complete the Arena questline and at after you do this then go to sleep in a bed and you should be someone should come to you and give you a quest to join the Dark Brotherhood.

After you sleep in the bed after the Arena questline and the side quest you should be arrived by someone after you sleep in a bed and he will give you a quest and after you finish the quest then you will become a member of the Dark Brotherhood and do the quest he tells you to go the Inn Of Ill Omen it is a Inn in it is north of Bravil and south of the Imperial City and it will be marked on you're map for you to check and there is someone there called Rufio and you need to kill the person in the Inn to complete the mission and Rufio he is usally sleeping in the lower rooms of the Inn and he usally sleeps and you can quietly ambush him while he is sleeping and then leave the building and usally no one will see you do that there is no one else down there on the lower floors of the Inn I think and now you need to go to a bed and go to sleep again.

After you do the mission and go to sleep the person will come back to you and he will tell you that you are now a member of the Dark Brotherhood and he will tell you where the place of the Dark Brotherhood is it is in a secret location in Cheydinal you need to go through a abandoned house and into lower floors and there will be an evil door that you need to pass and it will ask you a question and you can answer it correctly the person who talked to you after sleeping tells you how to enter what the answer of the question is and after you go through the door there is someone waiting for you and you will be given Dark Brotherhood armor and now you have joined the Dark Brotherhood now and thanks for reading my guide.