On the physical side of things, try plugging into the headphone/line-out. If that's occupied, many motherboards have an on-board header for audio - you can pull it from there as well (if you're installing something in a case that would actually be cleaner).
Where driving the spectrum analyzer is concerned, what you want is a Fast Fourier Transform (FFT). More info here:http://en.wikipedia.org/wiki/Fast_Fourier_transformhttp://en.wikipedia.org/wiki/Spectrum_analyzer
The little Atmel we use will actually do a decent FFT, and if you Google you can find example code + libraries for this. What you'll do is use the FFT to separate the signal into frequency bins, then use the magnitudes of those bins as inputs into your LED driver code.
Here is a Japanese guy who has done exactly this - but he's outputting to a display rather than LEDs:http://elm-chan.org/works/rsm/report_e.html
Take a look at his code though. He claims to be running his FFT + display code at 60Hz which is plenty fast for a spectrum analyzer.