Definitely DO NOT run the power cable through the wall. If it causes a fire, you'll not only lose your home, but you'll get zip from the insurance company. You can just plug the TV into a surge protector on the floor and use one of those plastic covers over it (example).How do you run wire through walls?
cut a hole for the new electrical box. Also cut a hole near the floor outlet to make the job of pushing the wire through easier. Start pushing the wire through the highest point in the wall and work your way down. Feed the wire and cables through the holes until everything is in place.