First floor has the cable modem, router, computer1. I was thinking I could use these jacks for the 2 wallmounts and use about a 50 ft rj cable to wire the 2 wall mounts together. Then I would use a standard rj45 cable from the router to wall mount on the first floor, and another from the 3rd floor wallmount to computer2. Would that work to give my second computer a connection? The wallmount holes/plates already exist I would just repace them (or use a 2 hole plate) with RJ45 jacks instead of what they are (one wallmount has an F-pin (cable coax deal)on 3rd floor and the first floor has a phone jack mount )
A rough sketch of the setup

Parts and suggestions are welcome!
Thanks!
Crimson