You can purchase any video tuner / capture PCI card ['Alchemy TV' ('http://miglia.com/products/video/alchemytv/') and 'Televio' (http://www.televio.com/')], or the 'eyeTV 200' ['http://www.elgato.com'] and use the respective software.
If you want to play the game - still using a TV; as well as, record the action - you will require a TV coaxial splitter and two 75 ohm [RG-59] coaxial cables. Plug the existing XBox to TV cable into the splitters 'IN' connector, one of the new coaxial cables into an 'OUT' connector and to the TV. The remaining new coaxial cable goes into the remaining 'OUT' connector and to the PCI video capture / TV tuner card's or eyeTV's 200 'ANT' input connector.