I know nothing about xbox360's, but from a general PC perspective here's the problem:
- you can't just hook up a crossover cable and expect the xbox to get an ip address. your PC is not a DHCP server, and therefore will not allocate an ip address to your xbox. in this scenario, you need to allocate a static ip address to the xbox which matches your home network's ip scheme.
- now that the xbox360 has an ip address, you need to configure your PC to share your internet connection (Windows XP does not do that by default). I would suggest searching Microsoft's website for assistance if you do not know how to do this yourself, it is in your network settings.
- provided that the xbox live environment does not require any port forwarding, this should be all you need. If it does, you need to configure your linksys router to port forward as necessary.
I have assumed here that A. you can specify a static ip address on an xbox360, and I have no idea if that's the case. B. that someone else has actually done this successfully, C. xbox360 does not have wireless b/g access.
The easiest solution? run a network cable from your linksys router through your house to the xbox360. then forget everything I typed above.