Can I run data (e.g., for computer internet access) through my rg6 coaxial cable?
The adapters you would look for are called MoCA (Multimedia over Coax Alliance) adapter kits, but they are not very common, and therefore can be expensive.  Basically the MoCA adapter works like a modem that connects computers over a telephone line, but it uses Coax instead.  Each computer needs an adapter, and the computer treats the adapter like any other Ethernet connection.

This PC World article talks about a MoCA installation.  Some of the important takeaways is that Powerline adapters can actually be faster, but MoCA might be more reliable.  If you have satalite TV on the Coax, MoCA might not be an option, and the reliability and speed of either one will depend on the wiring quality in the house.