I'd say get a monitor, VGA cords for the Xbox and wireless headphones/a cheap pair of computer speakers for sound.
I'm using a Samsung 19" monitor and one like mine is around $215 after MIR on NewEgg.
http://www.newegg.com/Product/Product.aspx?Item=N82E16824001094
Nice thing about that one is it has both DVI and VGA ports so you could hook both your 360 and computer to it at the same time.
Don't cheap out on the VGA cord, get the offical Microsoft one. Try getting it used from Ebgames/Gamestop. I heard they have them cheap.
For sound, I would say go with both the wireless headphones and the cheap computer speakers if you can. That way depending on the situation, you can just swap out the cables depending on the situation (they both use red/white so it's as easy as plugging them into the couplers)
Just about any pair of computer speakers should work as long as you buy a pair of couplers from Radioshack. Those will cost you less than $5. You can get a cheap 2.1 setup from NewEgg for $45+
http://www.newegg.com/Product/Product.aspx?Item=N82E16836116024
Wireless headphones are a little more tricky but this way, you can just plug your iPod into the 360 and stream it from there (or if you have something like this
http://www.newegg.com/Product/Product.aspx?Item=N82E16855995028, I'm pretty sure you could hook it right into the headphones)
I have these hooked up to my computer/360 and they kick ass.
http://www.amazon.com/Sennheiser-RS120-Wireless-Headphones-Charging/dp/B0001FTVEK
Just don't set the reciever close to the monitor or anything like that. It will pick up interference.
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Whichever way you go, it shouldn't be too much out of your pocket to get a pretty bitchin' set-up
Just be careful having all of that crap in your dormroom, you wouldn't want anything to get stolen.