Couple things to know does the phone wire go to a smart panel inside [preferably with a power port available], or does it go to a phone NID outside? If the ladder, so much for the easy way making it an Ethernet adapter. assuming its not really old and daisy chained making it rather pointless to try. I will assume there are 4 pairs on the wire and not red, green, yellow, and black otherwise forget the phone wire. Do you know its at least cat5e [not just cat5] or cat6/6a? Are you any good with hand tools? Because your going to have to do some hand work. Have you ever used phone connectors ? Any kind screw or IDC [insulator displacement connectors aka punch down]?
If you are sure you have cat 5e plus and it goes to a valid place, to use this Ethernet wish of yours, its as easy as buying a replacement plate [I prefer IDC , not screws but you can find both] and switching them out with the correct tools [punch down and snips tool or stripper, snips, and and screwdriver]. If not, the moca is a valid option, do you have any tv services on either of the outlets you want to use, if not terminators on the end and a DMM/VOM on resistance can locate a wire rather easily, splice the two wires togather so they are not part of the other networks and yourr golder, if they do have other devices you will need valid splitters, a poe filter to prevent your shooting foreign signal into the other networks and impairing those services for potentially you and your neighbors. Then the correct equipment to use the coax path, moca adapters, Ethernet cables to feed the adapters, power, etc...
to advise you better we need more info about the phone wire and coax wire networks.