I have 2 computers: one running Windows XP with a wireless connection to the Internet, and one running Linux with an Ethernet port. Can I get the Linux computer to access the Internet by running a crossover Ethernet cable from the Ethernet port on the Windows one to the Ethernet port on the Linux one?
Normally one would do this between two Windows computers using Windows ICS.
Step one is to turn off firewalls temporarily on both machines. You have to do something with the IP address on of the machines (probably the client) perhaps set it to the same address as the first. Unfortunately I don't know much beyond this because my experience comes only from having problems with networking machines conventionally that have previously been used with Windows ICS.
Cat 5 will do for normal length connection.
If you get stuck, buy a router it's actually a rather better solution, at least compared with some flakiness with Windows ICS.