Line up a fluorescent light housing under the counter so it will illuminate whatever you need it to.
Drill screws through the screw holes in the light housing into the underside of the counter. Use screws short enough to not puncture the top of the counter.
Run the power cable to a nearby outlet. If the outlet is further away, drill a larger hole through a nearby wall or in the side of the counter to run the cord through so it is out of the way. Drill another hole near the outlet and pull the plug out so the light can then be plugged into the outlet.
Stick small LED light patches to the underside of the counter with adhesive and plug them in to provide additional decorative lighting as needed. The patches are light weight and the adhesive therefore holds them in place.