If you are using a standard dialup internet connection, without DSL or cable, I would start by assigning IP addresses to each computer. You will want to go to your Network settings and assign an IP and Subnet to one computer. These settings would be added to your TCP\IP settings on your Network Card, do not change the dial up adapter settings. Example : One computer could have an IP Address of 192.168.1.1 with a Subnet value of 255.255.255.0.
The other computer will have the same Subnet Value, but assign a new IP address, Example: 192.168.1.2.
I would also go into the computer identification of each computer and assign a workgroup name which would be the same on both PC's. Example: "HomeNetwork" or whatever you want to call your group.
After you have both computers seeing each other in Network Neighborhood (Entire Network) and inside your Group, you are ready to begin the internet connection settings.
Up to this point, This setup would be for a dial-up connection,if you have a router then the above would not apply to you at all.
I use a router with cable Internet, so I would need a little more detail on how you are connecting to help you.
I have not used this, but I believe Internet Connection Sharing would be the best method for a dialup connection\sharing, but someone else may give you more insight.
If you have DSL or Cable then there would be an entirely different configuration to achieve what you want. If this is not the configuration you have, then myself, or someone else could easily help you out. I would send a clarification on your internet connection.