How do I feed an ethernet cable through a doorway without damaging the door or wall?
You can buy flat Ethernet cables. I'm betting you could get one under the door and still allow it to be closed.

Another option is a powerline Ethernet adapter. This is a set of modules that plug into your power outlets and allow you to transmit network signals over them. You run a patch cable from the router to a module in the same room, then plug in a second module in the room the second PC is in.

Also, if the rooms are next to each other, look around ducts, pipes that might pass through, etc. You might get lucky and find a gap big enough to fish a cable through. 

Another option is to install low voltage boxes in the drywall with RJ45 jacks. Wire the jacks together, color to color, and use patch cables from the router to jack one, and from jack two to the second PC. Just leave them when you move and it will look like any other jack in the wall.

And lastly, have you considered WiFi?