Well, somewhere along the lines you have to have a modem. The modem is the thing that changes the computer signals into sound signals that can travel over phone lines, you have to have one to be talking to me right now!
It sounds like you've got an internal modem, though. Does your phone line run into your motherboard, or a PCI slot? If you don't know what that means, then just nevermind.
It's possible to use your computer to just reroute traffic through to your Router... Do you have another ethernet port on your computer? The port should look just like the one your phone line is going in to, but in a different spot on the back of the computer.
I'm thinking it might be possible to just reroute the traffic through to the Router. But I'm not sure if you have to do any complicated setup for that to work right, I haven't done it personally before.
Also: DSL is a different kind of internet service that uses traditional phone lines, like Dial Up. Only it does some funky frequency techniques so that you can use the phone AND the internet at the same time. At approximately cable speeds, too. You'll just have to ask your ISP if they offer DSL in your area, or if there are any others who do.
DSL is usually preferable for locations which are kind of far away from town. It uses the standard phone lines, so it's technically available almost anywhere.