For routine cartridge cleaning, you just hose it off with a garden hose. They make these inexpensive fixed size brass nozzles for garden hoses that seem to be perfect for cleaning.
If the filter pressure after cleaning is not returning to the normal "clean psi" level, then you need to do a more through cleaning. This should not happen all that frequently, many people only do this once a year. In any case, to do a more through cleaning, rinse the filter normally, then soak the cartridge in a mixture of TSP or automatic dishwasher detergent overnight, rinse the cartridge again and then reinstall in the filter. Only use real TSP, with phosphates. If you can't find that, use automatic dishwasher detergent. Use one cup TSP/detergent for every five gallons of water.
In rare cases, if the cartridge has calcium scaling, you can do an acid soak after the detergent soak. But this is not normally needed.
