"I want to use my Xbox with my computer, but I don't want to use the TV screen. I want to use the computer screen instead. Is there anything that I need to get to enable this?"
As long as your computer monitor has at least one VGA connection, you'll need to things to have a great Xbox 360 gaming experience. First you need an Xbox 360 VGA cable. Microsoft makes the Xbox 360 VGA HD AV Cable, which sells for about $40. There are also 3rd-party Xbox VGA cables for about $20. Ideally if your computer screen has two inputs, you can leave your computer connected to one, connect the Xbox 360 to the other, and then simply switch to the one you need.
The other thing you need is a stereo female RCA to mini-plug adapter, for connecting the audio out on your Xbox 360 to your computer speakers. If you have computer speakers with support for ToS Link you can use that for audio instead.