We have cable - which goes into a router which goes into Computer A - Such that if Computer A is not turned on, then the internet doesn't work for any of the other computers.
Now, let computer B be one of these computers which connects to the internet via Computer A/Router.
How would I connect another computer ... Computer C INTO computer B ... such that Computer C has internet aswell?
Computer B has Dual Gigabit lan, i think it "should" be possible but I have very little idea of how to do it. I assume it's not the blue cable because I'm trying that right now (Only left over cable I had).
You could probably ICS the ICS connection from the other computer but that would get hairy fast. I highly recommend either getting a new router with a switch built in or just getting a switch to plug your router into. Then plug the computers into the switch so all the computers can have internet at the same time and you dont have to worry about the others being on to get internet.
What he's saying is, what kind of router do you currently have?
You can pick up a Linksys router with a built in 4 port switch for between 40-60 before any rebates. This would make your setup very simple, allow all computers internet access independently and make your life easy.