I had to move the XBOX 360 downstairs to my office for about a week, so I had to figure out how to connect it to my second computer monitor (a Samsung with DVI inputs). After a little trial and error I finally got it working. Here are my three tips for getting it done.
Tip No. 1. Aim for the best picture possible
If your monitor takes a DVI input, then hook it up using the HDMI output of the XBOX 360. If all it will take is a VGA monitor connection then use that. There are adapter cables you can buy that will go from the XBOX 360 A/V connection to VGA for not very much money, so you should be able to do everything with one connection (audio and VGA for the video). This is the cable I used at first (with a VGA/DVI adapter) and the picture quality wasn't that great. It was passable though, and I used it for a few days because I had no other way to get the audio out of it. Audio? Read on.
Tip No. 2. It's all about the audio.
The problem with HDMI to DVI is that DVI doesn't take the audio from the HDMI signal. This makes sense from the computer monitor's perspective, because a monitor never has speakers anyway. So you're going to have to get the audio out from the A/V connector. But the problem is you won't be able to also use a standard XBOX 360 A/V cable because Microsoft (in their infinite wisdom) decided to put the HDMI port way too close to the A/V port. The audio will output out of the A/V port, but you got to fit that damn cable it. You can hack your cable and take the plastic housing off so it will fit, or you can buy an adapter that is specifically made for this purpose. (Actually, I think they are mostly used so people using an HDMI connection can also use stereo headphones, but it will work for your purposes as well.)
As for your speakers, you can use your computer speakers, stereo system or whatever you've got hanging around. Keep in mind you might need a male RCA to mini jack to make it work. If you're using optical audio, then you'll need one of those cables.
Tip No. 3. Don't ever overpay for cables.
If you plan ahead you can get this whole project done for about $20. However, if you're off to Best Buy to get the cables you need, expect to pay about five times more. Cables are cables - especially cables that transfer digital video. It's digital people. It either works, or it doesn't work. Gold plated, super gauge blah blah blah - none of that really matters. Don't pay more than a few bucks per cable. I use Monoprice.com because they have awesome prices, great quality and fast shipping. (The XBOX 360 Audio Adapter costs $20 at Best Buy, while at Monoprice.com it's $2.79.) I can't say enough good things about them.
My biggest problem now is, with the XBOX right next to my computer it's WAY to easy to play instead of getting work done. I'll get it back upstairs this weekend I swear.
Right after I kill that zombie.