It is very difficult to break into the NTSB as an investigator. They are chronically understaffed, and it is almost impossible with the current Chairwoman to get funding for additional investigators. You can watch the NTSB website for vacancy announcements and submit your package, but the competition is VERY stiff. In fact, in many cases, they have their candidate selected, THEN open the vacancy to the public, since by law, that is the way they have to do it...but then the vacancy closes, and their candidate is selected. The entry level investigators are typically in the various field offices doing GA investigations.
They also offer student intern programs, and this is one way to get a foot in the door at one of the field offices. However, there are only about 50 or so investigators nationwide, and vacancies are very, VERY rare.
I have pretty good and accurate info, since I am married to a former investigator, and have a fairly good insight into what goes on at the Board, since she still keeps in very close contact with many of her former co-workers there.
The FAA also maintains an office of accident investigation in DC, but there are only about 7 investigators in that office, so vacancies there are also extremely rare. The other ways to get into accident investigation are with the safety department at the various airlines, and aircraft and engine manufacturers, to name a few. But again, these are highly specialized positions, and again, are difficult to break into to. Accident investigation is like a close knit family. Everyone knows everyone else, and it is a tough nut to crack. The student intern program is probably the only way for someone without an actual investigative background, say from the military for example, to break into the field.