Assuming you are in the UK, and therefore talking about virgin cable services then the only way it can be done (legally) is by using the virgin set top box, outputting the video signal from that into a video or TV capture card on the PC and then use something like windows media centre as the recorder.
you can use a TV as the monitor as long as it has a suitable port for connecting the computer to it, most modern flat screen TVs have a standard VGA port on them so they can be connected to a computer no problems.