The only tool that you will need to purchase for this project that a typical homeowner will not have is a dryer lint brush. Most brushes will cost about $20-30 and can be purchased online or at any major home improvement store.
The first step is to clean the lint screen and the lint screen cavity. Pull the lint screen out of the cavity by tugging it straight out. Remove the excess lint from the screen with your hands before using a dryer vent brush to carefully clean it. Then, with the dryer vent brush, gently reach down into the cavity and spin it around to remove any lint. Remove lint from the brush using a vacuum. After the screen and cavity are clean, you can move on to cleaning the dryer vent.
Clean the dryer transition duct by sliding the brush through the duct. If your dryer is installed at an exterior wall and vents directly to the outside, the only other step is to clean the outside portion of the vent.
Some dryers have vents that extend 25 feet or more through the house walls, crawlspace, attic, etc. before terminating at the exterior. In that case, you will likely need to seek the services of a professional to clean the long section of the dryer vent. If you push the dryer vent brush as far as it will go and it does not reach the end of the vent, you are likely in this position.
You can find more information about this in the following YouTube video: