Solving all their quests and (i think, at least) oust Elder McNamerra from his position.
Complete Still in the Dark, find the missing laser pistol, and complete the quest that the Elder sends you to join the BoS (quest depends on who is Elder). That's really the only things to do to gain BoS fame.
adding on from the last, capture the virus or whatever it is in the archives area.
You get more reputation with BoS if you keep McNamerra on as Elder. You gain more when you get the parts to fix their Air Filtration system then you do when you oust him. Also if you have ED-E and you are at the point in ED-e My Love when the BoS and Followers contact you, bring it to BoS for Rep.