Since you do not have a dedicated router, one of the two PC's has to become the "router". Preferrably with two NIC's.
Ideally, it will be running an implementation of NAT, compatible WAN protocol (PPPoE, MPoA, PPP, etc.) depending on your type of internet connection.
Depending on which one your choose, off the top of my head, your Linux distro should have a how-to on their website on how to setup NAT.
Need a bit more details... Linux distro, Windows OS, hub or switch?, strictly TCP/IP, need MS file sharing as well?, etc.