Buy a cut-in receptacle at your local electrical supply store. These receptacles feature special clips that allow them to be mounted securely directly onto drywall.
Determine the location where you want the receptacle installed. Outline the hole for it using a pencil and a straightedge. Be sure that the location is not directly over a stud.
Cut out the hole with a drywall saw by poking the tip of the saw through the drywall then carefully making a saw cut on all four sides.
Feed the necessary wiring for the receptacle to the locating by pushing it up through the wall from the basement and pulling it out the hole you created.
Feed the wire through the back of the cut-in receptacle, then install the receptacle in the wall by pressing it through the hole until the clips engage against the back of the drywall.
Wire the outlet by attaching the bare wire to the green screw on the bottom of the outlet, the black wire to the gold screw on the side of the outlet, and the white wire to the silver screw on the side of the outlet. Press the outlet into the receptacle box and screw it on.
Place the wall plate over the outlet and screw it on by inserting a single screw through the hole in the center of the plate and into the hole in the center of the electrical outlet.