I've done that before, used to have dial up and two computers before I had cable. I can't quite remember exactly how we got it done but I believe it was just as setting up any other shared connection. Let me explain:
Go into the computer that has internet, go to Start, then Run, and type cmd, once in the command prompt type " ipconfig /all ", write down the numbers for IP Number, Gateway address, Subnet Mask and both DNS numbers. Go to Panel Control, Network Connections and see if there's a dialup connection listed, if there is, right click on it, go to Properties and there's going to be a list, in the list look for "Internet Protocol [TCP/IP]" click on it, then click on the Properties button under it. Once there click on automatically get IP address. Add the two DNS numbers to the bottom by clicking on 'use the following DNS...", also make sure that when you click on Properties for your network card [should be in the same place as your dialup connection icon in Network Connections] going into the advanced tab it has an option to share the connection, so should the dial up one, both have to be checked.
Now go to the other computer which is already set up in a network with your computer that has internet through the router, right? Go into Panel Control, Network Connections, look for the network card icon, right click it, go to Properties, go to "Internet Protocol [TCP/IP]", then properties and fill up the DNS numbers, if that doesn't work, then add also the subnet mask, gateway number and IP address but that shouldn't be necessary, if you have to do it though, change the last number of the IP number to a number higher than the one on the computer that has the internet, say if that computer has an IP 192.168.1.100, change yours to 192.168.1.101.