I would like to hook up the DVD player directly to the cable box. Do I have to also hook something to the TV. The TV is connected to the cable box with HDMI cable.

Depends on the DVD player.

You don't connect it to the cable box... you connect it to the TV.

Good: Composite (yellow for video; white and red for audio).
Better: Component (red, green, blue for video; white and red for audio).
Best: HDMI.