XDK (XBox Development Kit vs. Software Development Kit I believe) is the development tool. I don't know much about it nor do I know if it can be run off a standard xbox (there may be a difference between a development xbox and a standard). This sort of makes sense considering I've never seen any tutorials posted on using XDK. You might try to contact someone in the biz here - the Homebrew Forum, or one of the active coders or teams.
As far as for getting a video stream to the xbox, the only two digital methods are Firewire and DVI both of which have very limited support from the cable/sat companies and neither of which are supported by Xbox. My best guess is that you'd have to capture on a computer and then send to the Xbox to play (and capture again/transcode - if it can be done). I have zero confidence that this is going to do what you want and make the Xbox a PVR.
I'd post this again in the Homebrew forum (or maybe Xbox Media Player/Center) in the case that someone involved in development comes accross it (be much more clear about XDK in the title). I'm not confident that you will get the necessary traffic in this forum since it has almost nothing to do with XDK.