Let's start of by setting your expectations. There is no easy way to do this the right way. You will get wet and dirty. It is also going to take some time.
First you will need to drain your fountain. You can use your fountain pump to do this by redirecting the hose out of the fountain. Unplug your pump when it starts to take air in with water. Use a wet vacuum to get the remaining water from the bottom. If you do not have an easy to access pump, a bucket will work, but it will take forever.
Next, time to clean. Take our your pump. Clean it in a bucket of warm water with some dish soap. Remove anything big by hand. You will clean your fountain based upon what material it is made. If it is slate, clean it with a soap pad. If it is concrete, use a pressure washer with a fan tip. If you cannot rent a pressure washer, use half a cup of vinegar to one gallon of water and scrub everything. You could use two tablespoons of bleach in a gallon of water instead of the vinegar solution. Be sure to use waterproof gloves if handling bleach. The pump can be cleaned with either the vinegar or bleach solution. Be sure to thoroughly wash your fountain when finished so it is free of cleaning solution.
If there are any cracks, now is a good time to fix them. Let your fountain dry completely. Use caulk and sealant / paint to get your fountain looking like new. Allow it to dry to the time specified on the label.
Put your pump back in a fill your clean fountain back up. To avoid doing this for a while, keep your fountain maintained. Using a small fishing net is an easy way to keep leaves and twigs out so they won't dirty up your fountain. Use mosquito pellets to keep mosquitoes from breeding while allowing the water to still be safe for birds or pets.
Enjoy your clean fountain!