Whoa. What computer on the planet doesn't have a "line in" jack on it?? Unless I'm smoking some really wicked crack I'm thinking that even the built-in sound on motherboards has at least: line-in, line-out/speaker, and microphone in.
yeah, gotta agree with Jjones - either you are mistaken about your computer, or you have a really old piece of crap computer. I'd say, if the unlikely scenario you claim is true, you're only other option is to buy a new soundcard/audio interface.
If you need a new interface, there are a few USB or Firewire options that will set ya back about $100. M-Audio MobilePre, for instance
(just the one that I am familiar with - there are other options, too).
Or if you are not too concerned with the quality of your audio, just any old cheap soundcard - for gaming, etc. - that has a stereo line in could work.
But I would DEFINITELY take a closer look at your computer first to make sure.
Edit - And regarding the concept of doing it in two passes - I'd not bother. IT'll never sync up even close, and you'll spend forever trying to nudge the wavs around in your recording program, and still probably be disappointed. You really have to get both tracks at the same time.