I have an HP multimedia center computer and I would like to hook up my TV cable wire to it. Do I just split the cable connection with a connector and hook it directly to the back of the computer or do I need a cable box. I would like to watch and record with my computer.

Do you have a cable tuner card in this machine ?
If not you will not be able to hook the cable directly to your computer.
You may be able to run an areal line from the cable box to your computer, but you will need a TV capture card to be able to watch/record information from your cable TV.