This topic has been covered here and on the accompanying websites before; but it is so important it demands reiteration.
The simplest way to clean a slow sand filter is to use “wet harrowing”. Stop the output flow (put a plug in the output pipe so nothing can flow out while you are cleaning the filter), then agitate the water JUST ABOVE THE TOP OF THE SAND until the water becomes cloudy; taking care to not disturb the top of the sand. Then drain off the cloudy water by opening the wet harrow drain (the wet harrow drain is a drain valve right above the sand surface running outside the filter) to allow the cloudy water to drain off while keeping the sand surface covered with water by adding water to the filter input. Do this until the wet harrow drain water runs clear. Then close the wet harrow drain and open the output pipe and restart the filter. After wet harrowing, Let water run through the filter for at least 2 weeks before using the water. If the sand is slightly disturbed the filter will need more than 2 weeks to re-establish the biolayer (schmutzdecke) ( a coliform test might be a good idea if the sand is significantly disturbed.)
There seems to be a misconception that backwashing a slow sand filter is a good way to “clean” it. Nothing could be further from the truth. Parts of the problem may be a misunderstanding of the process of backwashing; and lots of individual ideas of the meaning of the word as applied to the process. For these purposes here, “backwashing” means forcing water back up through the gravel and sand bed until the entire contents is fluidized. Backwashing a slow sand filter, will totally destroy it. Here is a simplified explanation of why: Gravity and biological processes form layers in the sand. The gravel at the bottom keeps the sand from flowing into and clogging the output pipes. The top layers of sand should never be mixed down into the bottom layers because the top layers have lots of microbes that eat the bad stuff. We don’t want the bad stuff OR the microbes that eat the bad stuff in the output water. The layers formed biologically and physically in the process of water flowing slowly DOWN through the sand keep the bad stuff and the helpful microbes away from the output water. Backwashing will mix all this stuff together and give toxic output; and, screw up the gravel layer at the bottom permanently destroying the filter’s operation. These filters don’t work with gravel and sand mixed all through the bed. There are some biological filters that have a modified form of backwashing used to break up the schmutzdecke, but they are carefully designed and not practical for a DIY project.