I worked in an engineering firm who were frequently contracted in to provide analysis for accident investigations.
I was asking one of their investigators about how to get into the AAIB and it is pretty much as Ghengis says, their website only gives details of requirements when vacancies are available but may be worth keeping an eye on it.
I'll say this much though, incredibly interesting work but very morbid. Seeing the Pan Am 747 Forward fuse still built on one side of the main hangar and wandering around an aircraft that looks more like screwed up tin foil than an aircraft knowing that four people at some point were in it when it turned to tin foil. Very thought provoking and harrowing.
Before flying, the investigator I worked with would wander around the hangar to solidify the importance of safety.