If you're on a budget (who isn't), then the xbox 360 is NOT the way to go. In order for you to get it working you would need a 360 ($400 USD or more), Windows XP MCE ($125 USD) and a supported capture card ($75 USD), plus who knows what other extras plus taxes. You would be $600 USD minimum, and I bet it would go higher.
A better solution would be to mod an original xbox and install XBMC. You should be able to do that for less than $200 USD.