Routing an ethernet cable - eviltoast

Hey everyone!

Looking for advice, so, I’ll keep this short.

I want to run an ethernet cable from my living room (from the router) to the bedroom directly above. Unfortunately drilling holes in the walls isn’t an option as It is a rented house.

I currently use home plugs which work okay, but I don’t get the speeds I need.

Is it feasable to cut an ethernet cable sized hole in a wall outlet, and run the ethernet cable to the outlet directly above, in which I would also cut a cable sized hole for it to fit through? It seems like it would be straight forward, but I don’t want to get electrocuted haha. And I’m aware their will be a much better way to do this.

Any advice?

Thanks