An FFT is not going to get you the information you're seeking. It gives the frequency spectrum of a time varying signal, not the tempo. If you try to sample the signal with a slow clock, you'll get massive alising, which will require a very low pass filter to remove. If, in fact, you do such filtering, you could concievely use FFT to obtain the frequency of the beat, as it is really just a low frequecy periodic signal.
You might try "peaking" the music or low pass filtering, and see what you get. Since the beat can be changed or syncopized ( ie, alternated between upbeat and downbeat) you'll need some pretty intellignet software to sort it out. I do know there is inexpensive software that does some of the work for you. Not sure what you're trying to do, but something like "The Amazing Slow Downer" might be a start. Might spend a few minutes researching that.
Edit: After thinking more about this, I added the underlined part. It seems to contradict my earlier comments, but reflects a modified conclusion after more consideration.