Ethernet has a way to run over coaxial cable.
It's called 10-Base2, and you can typically find a few PCI cards around now that still have it handy. Perhaps an old 10-Base ethernet _hub_ will have a port that it works with.
You will need to get special cables made to go into the F-Type connectors on the wall jacks used for typical cable tv installations, because 10-Base2 uses locking J-Type connectors (IIRC, which I probably don't because it has been a while, but the 10-Base2 connectors definitely lock). Then get a run that goes into a "T" connector and a terminator for both your computer, and the hub that is plugged into the router.
There may be a problem with the cable itself, because 10-Base2 and Cable TV cable run at different impedences, so you may very well get no signal at all.
http://www.windowsnetworking.com/art...ials/thin.html
There is the best guide I could find.
Here is an ethernet card (You should be able to find one at any used computer store for less than $5, but this is what I found online):
http://www.starmicrotech.com/productpage.cfm?prodid=58
And here is a hub (Sadly, it is cheaper than the card I found online):
http://store.yahoo.com/tcoutlet/10b9porhubpl.html