Adding a second Router? Coax to Cat5?

I have a room in my house that only has wired access to my network through a coaxial cable (No Cat5 and wifi doesn't reach it). I would like to add a second router to that room in order to enable wired access via Cat5 while also increasing wifi signal strength.

Can I simply add any router to the system? Or is there a better solution to convert that coaxial ouput to a Cat5?

As you can tell I don't exactly know what I'm doing, so I appreciate the help.

Thanks.