Start by turning off the power to this circuit. Then, remove the outlet cover plate and pull the outlet out of the metal box. Take a look and see what side of the box the wall stud is on. You'll want to locate the new outlet on the same stud and on the same side, to minimize any drywall patching. Essentially this means the new outlet will be directly above the current one. You should be able to remove the metal box - it is often fastened with a few screws that you can access through the front of the box. If not, cut 3" out of the drywall on top of the outlet, and also cut a new hole in the drywall for the new outlet. (Make it the same size as the opening around the current outlet.) Now you will be able to feed a wire down through the wall. Tie the wiring into the existing circuit. It would be easiest to leave the original outlet in place - you're not supposed to make wiring connections that are inaccessible, so don't remove the outlet and cover the hole with drywall! If you don't want to keep the lower outlet, then put a solid cover plate over the box.
Remember that all electrical work should be checked by an electrician! Be safe.