Basically what you're going to be getting is an outboard sound card, and a DI box. (Assuming you already have the guitar and headphones ? ) Some soundcards are intended for use with a guitar and don't require a DI box. The main reason you can't just use your PC's onboard sound card is that it's too slow to do real time guitar - you need to get something with low latency. This is where I hand over to other forum members to suggest a good cheap sound card... I'm using the Guitar Rig Kontrol, which is great 'cos it doubles up as a pedal board and you can control everything with your feet, but you will undoubtedly get something which is just as good for cheaper (if you don't need the pedal board). [Edit - That Behringer Guitar Link looks decent!]
Once you've got that running, I'd say download the Guitar Rig demo to get acquainted with playing through the PC. You simply plug in all your gear, start up Guitar Rig, select your amp and play. You're done! (for now)
Guitar Rig is great 'cos it has a lot of amps and effects all in one package, with an easy to use interface. Unfortunately it's starting to fall behind, and there are free amp sims available which sound better. So before you spend money on the full version, check out this site:
http://www.guitarampmodeling.com/ . You'll find links to all kinds of amp sims, most of which run as VSTs (Virtual Studio Technology), which are plugins that you need to load into a DAW (Digital Audio Workstation) like Reaper (free unlimited demo at
http://www.reaper.fm/). (That sentence was a mouth full - don't get scared off!). A DAW is overkill for what you want to do, but really is the simplest way to run VSTs. Plus, if you get this far, you've got everything you need to start making some high quality recordings - bet you weren't expecting that!