Unplug the dryer from the electrical outlet and pull the dryer away from rear wall so you can access the exhaust port. Loosen the screw on the clamp securing the dryer duct work to the back of the dryer, using a Phillips-head screwdriver. Pull the ductwork away from the dryer.
Disconnect the duct from the exterior vent in the same manner as you did in Step 1. Push a dryer brush through the pipe of the exterior dryer vent and rotate the brush. Pull the brush out, removing any lint that you didn't push out of the other end.
Insert the end of the electrician's fish tape into the lowest end of the dryer duct. Most ducts within the wall travel up from the dryer. Continue feeding the electrician's tape through the duct until it comes out on the opposite end. If necessary, have a helper on the other end to let you know when the electrician's tape has come through.
Attach the handle of the dryer lint brush to the hook on the end of the electrician's tape. Slowly crank the handle on the electrician's tape reel to pull the tape through the ductwork. Remove the lint from the end of the ductwork as it falls through.
Continue cranking the tape reel handle clockwise slowly. When you feel some resistance, stop cranking. Grab the tape with your fingers and push up and down slowly to unjam the dryer brush. The resistance is most likely due to bending around a turn in the duct or a buildup of lint.
Start cranking the reel handle once you move past the resistance. Continue reeling in the tape until the brush comes out of the ductwork on your end. Unhook the brush from the electrician's tape and insert the brush into the dryer exhaust port. Remove the lint from the exhaust with the brush.
Reconnect the ends of the ductwork to the exterior dryer vent and to the exhaust port on the rear of the dryer. Tighten the clamp screws to secure the ductwork. Push the dryer back toward the wall and plug in the power cord.