I'm doing basically the same thing. I was going to hook up an XBOX360 VGA cable to an adapter for DVI, but ran across a better soution for about the same price!
(If you do not have an XBOX360 with an HDMI port, scroll down to "NO HDMI PORT")
I have an Acer AL2216W monitor that has a DVI-D output. I am getting a Dynex 6.5 Ft DVI-D to HDMI Cable from eBay (Under $10! Dynex model DX-C114181). Best Buy sells the same cable for $59.99, so eBay is probably the best option. Video should be better, but you will lose sound because HDMI carries sound, but DVI does not.
2 Solutions:
1 - Buy Optical / RCA Audio Adapter
or the cheaper way,
2 - Take the RCA cable (one with Yellow, Red, and White) that came with your xbox and open the casing where it plugs into the 360 with a small flat tip screwdriver by prying it apart. (You have to remove the casing so there will be room for the HDMI cable). It may get a few nicks from the screwdriver, but will snap back to gether if you want it to later. Once you get the casing off, put it somewhere you will remember. You won't need it. If you have the one with six RCA plugs and opical, the connector will be loose. Use electrical tape around it and it will be more secure. I suggest taping it up even if it's not loose. With the casing removed, you'll be able to plug in the HDMI port. You will only use the RED and WHITE RCA for the sound.
This will only work if you have a 360 with an HDMI port.
NO HDMI PORT
If you don't have an HDMI port, the best solution is to find out what type of DVI input your monitor has. More than likely, it will be DVI-D. Search on eBay for a DVI-D to VGA Adapter. You'll have to have the XBOX360 VGA cord and make sure the adapter you get has the correct ends (ie male or female). Don't just buy a DVI to VGA cable. I found one on ebay for like $1 with $0.57 shipping and it didn't work at all! Although this SHOULD work, I have not personally tried it, but I do know they make a DVI-D to VGA adapter that should do what I just described.
Hope this helps!