Holden Karnofsky of Givewell discusses some of this in his EA summit talk. There's no simple answer, but a short one is "get big". Near as I can tell, the best way to do this is develop rare and valuable skills that interest you, a la So Good They Can't Ignore You. Personally, I think math and computer science are good places to start. Both are rare and valuable (especially taken together). If you have aptitude and interest (as I estimate you do), start there. For math, step 1 is to get through calculus. You'll get different opinions for CS; I'm personally a fan of SICP, but that assumes calculus. Fortunately, we've compiled a list of programming resources. And then things that strike your interest. I'm learning psychology, writing, and economics, not because I think they're the rarest or most valuable skills, but because they're at least somewhat uncommon and at least somewhat valuable and I really enjoy learning them, and the combination of math/CS/psych/writing/econ is sufficiently novel that I should be able to do useful things that wouldn't happen otherwise. Holden discusses reasons for choosing things that interest you/things you have aptitude for, rather than the most tractable problem, in the video linked above. Good luck!

If you look at the give well job description for a research analyst (http://www.givewell.org/about/jobs/research-analyst) it doesn't mention that GiveWell is interested that people who apply have a degree. If that's where you want to go applying directly to GiveWell would be the straightforward course of action. Given FHI's academic nature they probably do prefer people with degrees but I think FHI doesn't want specific degrees but wants to hire people with expertise that they currently don't have so they should be pretty open.