What is my best bet for recording game play on the 360?

I have my Xbox 360 hooked up to my 1080p HDTV via HDMI. I want to record game play in the best quality possible but I can't spend much more then $20. Could I hook up an HDMI or VGA cable from my tv to my computer and use video capture software, or would it just show the computer screen on my tv?Any help would be appreciated!