Turn off the electricity to the garage outlet that you're going to wire the floodlight into. Test the outlet with a voltage tester to make sure the circuit is dead. If you get a reading, switch off the proper circuit and test the outlet again.
Unscrew the plate cover from the outlet, and then remove the mounting screws that hold the outlet in the electrical box. Loosen the terminal screws on the side of the outlet, and disconnect the wires.
Screw the mounting plate for the extension box to the electrical box you removed the outlet from.
Punch out the knockout hole from the top of the extension box, and connect a 1/2-inch metal conduit connector to it. Screw the extension box onto the mounting plate. The extension box is similar to a junction box -- it will protect the electrical connections between the existing cable and the one that will run to the junction box.
Measure the distance between the ceiling and the top of the extension box. Cut with a hacksaw a piece of metal conduit that is 1 1/2 inches shorter than that distance you measured, and file rough edges off of the ends of the conduit.
Insert one end of the conduit into a right-angle connector and the other end into a conduit connector on top of the extension box. Hold a level against the side of the conduit to check it for plumb (vertically straight), and secure it to the wall with conduit hanger. Turn the conduit so that the open end of the right-angle connector faces the direction of the floodlight.
Slide a section of conduit into the open end of the right-angle connector, and secure it to the wall with a conduit hanger every 12 inches. Install enough conduit to reach the location where you're installing the floodlight, using straight couplings to connect different sections of conduit. Use a conduit elbow if you need to go around a corner.
Drill a hole through the exterior wall with a 7/8-inch spade bit over the end of the conduit. Place the hole 1 inch beneath the ceiling.
Connect the junction box to the end of the conduit, and then screw the box to the ceiling.
Slide 14/2 cable through the junction box to the outside. The 14/2 indicates that the cable contains 14 gauge wiring and that there are two wires and a grounding wire inside the cable. Go outside and locate the cable.
Connect a cable connector to the outlet box for the floodlight. Slide the cable coming from the wall through the outlet box, and then tighten the screw on the cable connector. This prevents the wire from sliding around; otherwise, it the insulation could wear out and expose the wiring.
Apply a thick bead of silicone caulk around the hole in the wall where the cable emerges, which will prevent leaks. Place the outlet box against the wall, and attach it with 2-inch-long galvanized decking screws. Go back to the garage.
Remove the cover plate from the right-angle conduit connector. Slide electrician's fish tape down through the conduit and into the extension box. Tape 14/2 cable to the end of the wire from the fish tape, and then rewind the fish tape to pull the wiring up through the conduit. Pull the end of the wiring out of the conduit connector, and remove the tape.
Slide the fish tape through the conduit from the junction box to the conduit connector. Reattach the cable to the wire from the fish tape, and pull the cable through the conduit to the junction box. Detach the fish tape, and replace the cover plate on the right-angle connector.
Assemble the floodlight, following the manufacturer's instructions, and take the floodlight outside.
Cut the cable leading from the wall to about 6 inches long. Remove 2 inches of sheathing from the electrical cable, and use wire strippers to strip about 1/2 inch of insulation from the individual wires inside the cable.
Connect the wires to the floodlight. In many cases, you'll connect black wires together, white wires together and screw the green wire to a grounding screw on the floodlight, but read the instructions for the floodlight you purchased for the correct procedure.
Attach the floodlight to the outlet box on the wall, and screw the light bulbs into the sockets. Return to the garage.
Connect the wires from the floodlight to the wires from the extension box with plastic connector caps. Tuck the wires inside the junction box, and install the cover onto the box. Match the wires by color when connecting them.
Connect the wires inside the extension box, just as you did the wires in the junction box, and place the cover on the extension box. Turn on the power to the circuit at your home's main electrical box.