Remove the outlet cover by unscrewing the small screw in the middle of the outlet. This will give you access to the outlet itself.
Chisel out the grout from around the tile that has the outlet currently in it using the corner edge of the putty knife. If you're careful to avoid breaking the tile, you'll be able to reuse it, simply switching the outlet tile for the tile where the outlet will be located.
Turn off the electrical power in the area in which you are working. This can be done at the main fuse panel of your home. If it is a newer home, the fuse will be labeled. To turn it off, all you have to do is switch the fuse from "On" to "Off."
Pull the tile gently out of the grout bed into which it is set and then determine where the stud is located to which the electrical box is attached. Studs are generally located on 16-inch centers. To find the tile behind which the stud you need is located, measure along the wall toward where you want the outlet located in 16-inch increments. A stud finder device can aid in this. Once you find where the stud is at, remove the tile where you intend to install the outlet in the same fashion as you removed the first tile.
Remove the screws that hold the electrical outlet in place inside the electrical box, then pull the outlet out of the box to give you access to the electrical conduit. To remove the outlet, Loosen the screws that hold the conduit onto the wire and the wire will simply come loose.
Cut an access hole in the drywall underneath the tile for the installation of the outlet electrical box in the new location that is the size of the electrical box in the old location. Use a safety knife to cut through the drywall and then remove the outlet electrical box from the old location by unscrewing it from the stud to which it is attached. As with the tile itself, if you remove the old one carefully, it can be reused.
Route the electrical wire from the old location to the new location. You may find that you have to do this by drilling a hole through a stud in between the new and old locations. To accomplish this, use an electric drill fitted with a long-reach drill bit. The bit will allow you to drill into the stud at an angle so that you don't have to remove more tile than is necessary.
Straighten a wire coat hanger and then feed it through the hole you drilled so that you can reach it through both holes in the tile wall. Loop the end around that will be connected to the electrical socket to create the eye of a needle into which the electrical conduit can be threaded. Bend the conduit down tightly so that it will fit through the access hole you drilled, then pull it through the hole and toward the new outlet location.
Pull the electrical conduit out of the wall at the new outlet location, then install the electrical box by screwing it to the stud adjacent to the outlet hole.
Attach the electrical conduit wires to the outlet, then mount the outlet into the electrical box. Once this is done, reinstall the tiles. The tile that held the outlet should now be located in the new outlet location, while the solid tile should be located where the outlet had previously been located. Grout them into place, sponge off the excess, then allow the grout to dry completely before proceeding. You can, however, turn the electrical power back on at this point.
Install the outlet cover onto the electrical outlet to complete the installation process.