If the fan filter is a traditional aluminum filter, you can always use your dishwasher for cleaning it. Simply put the filter in the bottom rack of your dishwasher. Run a gentle cycle with your dishes. If you do not have a dishwasher, you can wash it manually in the kitchen sink. Rinse the filter with hot water and scrub it with a scrubber and dish soap to clean all the way in. Filters that have springs are recommended to be hand washed.
A good way to clean the filters is to use a degreaser. Especially for very dirty and greasy filters. First take out the filter and rinse it with hot water to remove some grease. Then cover it with degreaser. Different brand degreasers take different time for deep cleaning, so follow the instructions given for them. Once the grease has dissolved, wipe off the degreaser with a soft brush and rinse it thoroughly with kitchen sprayer. That is how you deep clean an exhaust fan filter.