If you have a stereo sound card, you should be able to plug a guitar directly into your computer, and record tracks into some audio recording software. I did a Google search on "audio recording software" and found free downloads, such as "Audacity" --
http://audacity.sourceforge.net/. I've never used Audacity, or any of the other free audio recording software in the Google search results (I'm using Cakewalk Sonar 8.5 Producer, which I bought when I ordered my audio recording computer from PCAudioLabs), so I cannot recommend any particular free software, but with a little research, I think you'll find something you could use. Apple's Garageband comes to mind, too, but again, I've never used it.
If you want the amped sound, get yourself a used Shure SM-57, put the 57 on your amp (play around with the placement to get the best result) and plug that into your soundcard.
Right now on Ebay, I see a seller with new SM-57s for $80, without the packaging. I see a used one just closed at around $71, I see another used 57 with 4 days left, now at $34.
57s are great mics, extremely durable, and very good for mic-ing guitar amps and snare drums. They're good all-around mics, useful for many different situations, including vocals. Another good mic to have is the SM-58, which are a little more pricey than 57s, but still affordable mics, and great for recording vocals on a budget.
NOTE: 57s and 58s use an XLR cable (female end into mic, male into soundcard/mixing board), so you'd have to get a converter for XLR to 1/8" mono male phone jack (mini-plug) such as this, which are around $5 to $15 (at
http://www.fullcompass.com):
So, for nothing, or less than $100-$200, if you get a mic or two, I think you could be recording to your computer.