First you talk to the head of security in the first room on the left when you enter the vault. He tells you to read the archives and talk to the computer guy down in the archives. Once you do that, you have to fix a computer virus by either having like 70 science or by doing a really, really annoying little mini-quest. After thats done, you read through the archives on the computer and you learn about the Chain of Command or something similar and you learn that the elder has broken a few rules. Go back to Hardin and he'll tell you to come back in a few days when he gets the elder removed. Leave the vault and wait a few days or go do some other quests or w/e, then return to the vault and a BoS scout should run up to you and tell you how much better the BoS is now and blah blah. Go talk to the former elder (who is still in the bunker in the same spot but is no longer in charge) and he'll give you a quest to join the BoS if you so choose. Finish that quest and the elder will give you the choice to be a paladin or a scribe in the BoS and he'll give you a set of power armor and access to the BoS shop.
Its something like this, I forget exactly but this is basically it.