Well. First of all Doctors generally don’t need to do this, usually you get this done from a nutritionist or personal trainer. But if you get your body fat percentage checked from a Doctor they would probably use the same tools:
This is the most common method. The doctor will check various parts of your body by pinching the fat in that area and using a simple handheld caliper measure the thickness. This is almost something you can do yourself but you need to understand how exactly to measure and where exactly to measure.
For this more expensive method the doctor will need to have a special water tank. They will calculate your weight before you go in the water and then measure that number against your weight while completely underwater. This method is very accurate.
This method is also more expensive and rare. The doctor will need to have a special body pod that seals off and calculates your body fat by measuring the air pressure and density of your body.
There’s some new devices and technology that will allow you to simply stand infront of a screen and allow it to scan your body. The computer can then calculate your body fat percentage automatically. If your doctor has one of these machines then he won’t need to do much work.