I have a very similar set up to yours, and mine works perfectly, here's what I use:
MX602A mixer (older version of yours)
I use channel 1 for the mic (Marshall MXL 2003 condenser) with phantom power. Also, I think you have to use an XLR cable for this
I use channel 2 for guitar with a standard patch cord
then my mixer is hooked up to the computer via RCA cables (the red and white ones, like on TVs, VCRs etc.) from L and R "tape out" to the computer. The cable I use has the L and R RCA on one end, and on the other end is a standard 1/8" jack which fits into the LINE in (NOT MIC IN) of your PC. This cable was picked up at Radio Shack.
Like others have mentioned before, you need to connect into the LINE IN on your PC, so make sure your computer is set up for line in recording, and not mic in.
That works for me, so hopefully it helps