I take it that the coaxil line is the cable input for the cable modem. That's not ethernet at all. You can split it, but to use the internet access from that cable will require another cable modem and another internet account with the cable company. Probably not what you want.

If you want to run ethernet, use unshielded twisted-pair (UTP) catagory 5 or better cable. There do exist two types of coaxil cable for ethernet, but both are seriousely outdated. The other option is wireless, of course.

Routers to share internet access are dirt-cheap now, and not too much more expensive if they also contain a wireless access point.