Ok, your issue is a little different since you are connecting the PC and the 360 directly instead of using a modem/ router with DHCP (automatic IP address configuration). You have DHCP on your router to the PC, but not from the PC to the 360. If you were to connect directly to the router either with a cable or with wireless, then you'd be sorted.
What you'll need to do is enable Internet Connection Sharing, to share the wireless internet on your PC with the network cable to your 360.
Unfortunately, I haven't done it in Vista, but I can get you a good start.
This thread is here to help with XP, for now. Have a look at the post explaining ICS connections, and it will explain the concept. The only issue is finding those same options in Vista.
We also have a member here, ShadowmanUK, who runs a site called
http://www.xboxlivetheguide.co.uk. I haven't looked, but he may well have updated his site with where things are located in Vista.
Doing a search in Google for Vista ICS might bring up some results for you as well.