The easiest, least technically demanding thing is to buy a router that you would connect both your computers to. That way you both share the same internet connection all the time. The best, but perhaps more technically difficult but cheapest thing to do is get an old discarded computer (an old Pentium 100 will do), put Linux on it, and have it do the packet filtering (firewalling), NAT (routing) and caching (use squid) for both machines. (You will also need an inexpensive hub or switch.) This is what I've done at home. I have 6 machines sharing a single dsl connection. There are HOWTO's on this at many places on the web.
Where are we going? And what am I doing in this handbasket?