All you need will cost you as little as $35 dollars and some patience:
1) Amp with a headphone/line-out jack (you probably have this already)
2) Soundcard on your PC with a line-in port (you almost definitely have this already)
3) 1/8" to 1/8" stereo mini plug cable ($5), to connect amp to your pc
4) A pair of big headphones (at least $30) that come with the 1/4" big plug adapter
5) Audacity.. free pc recording software,
http://audacity.sourceforge.net
You can get all of this at any mainstream major electronics store. The software is free. I assume you got a pc, not a mac.
Stick the 1/4" adapter on one end of the stereo cable, now plug it into your amp's headphone jack.. then plug the skinny end into your pc's sound input. Install audacity on your pc. Now, to make sure the software "hears" your amp, double-click on the volume controls icon in your system tray and make sure the line-in is not muted.. then while still in the volume controls go to OPTIONS, then PROPERTIES, and show RECORDING CONTROLS and make sure the line-in is checked.
Also, you can keep your gain up for distortion, but keep your volume low on the amp.
Plug your headphones into your soundcard where speakers would normally go so you can hear everything.
Mess with the software a little bit to figure out how to record.. usually very easy.