The cheapest (but least flexible) way to achieve what you want to do is to connect your amplifier's headphone jack to the line in of your computer soundcard. Radio Shack or the Source (even Walmart) should carry the appropriated cable.
As slippin mentioned, a mixer would be the ideal way to go. But if you have an iPad or even an iPhone, you can buy a device which permits you to connect your guitar and earphones to it.
Its a small device that has a phono jack for your guitar, and a mini jack for your headphones. You then use apps such as Garageband or Amplitube as your amp and effects rig. Of course this is not your amp, but you can get some pretty decent sounds (especially with Garageband).
You can buy a host of apps, that do different things.. even an 8-track recorder.
Here are some links you may find interesting:
http://www.ikmultimedia.com/products/amplitubeipad/
https://itunes.apple.com/ca/app/garageband/id408709785?mt=8
Another cool option is to go with a pod from Line 6. This has all the sounds you want and more (and it has a headphone jack).
http://line6.com/podhd/desktop/index.html
Nice thing about the pod, is that you can use it with the amp as an effects processor when the wife and kids are out of the house.
Hope this helps, feel free to ask for more details.
Izzy
