If you have a cartridge filter, you can clean it in trisodium phosphate (TSP) or dishwasher detergent (surfactants such as linear alkyl benzene sulfonates) as described in
Cleaning a Cartridge Filter. However, if you don't see a rise in pressure, you probably don't need to clean the filter (unless it's oversized and it's been a long time, such as an entire season).
(bik1906, you can skip the following; it's clarification of what chlorine can and cannot break down chemically)
As seen in
Reactions of chlorine with inorganic and organic compounds during water treatment -- Kinetics and mechanisms: A critical review, chlorine reactivity with organic compounds is limited to particular sites (mainly amines, reduced sulfur moieties or activated aromatic systems). In practice, chlorine reacts very quickly with ammonia (though it takes up to several hours to fully oxidize the resulting monochloramine to nitrogen gas), sulfur-containing amino acids (cysteine and methionine), primary amine amino acids (glycine, alanine, serine, valine), secondary amine amino acids (sarcosine, proline), peptides (only those with terminal amines), parts of some proteins (mostly sulfur-containing), methyl ketones or aldehydes and special cases such as citric acid. It does not, in general, react either at all or very quickly with triglycerides that are a typical component of vegetable oil and many greases nor does it oxidize fatty acids.
In short, if you don't see a
nitrogen (especially amines, so no double-bonded oxygen in the carbon next to the nitrogen) and you don't see a
sulfur and you don't see a double-bonded oxygen next to an end carbon, then odds are that the organic isn't going to get broken down by chlorine.
In an outdoor pool, the breakdown of chlorine by the UV in sunlight produces hydroxyl radicals that are very powerful oxidizers and these can oxidize many organic compounds including greases and oils, but this is somewhat limited in its capacity since the radicals are very short-lived. Even so, such oxidation is not necessarily complete and may just make such compounds more water-soluble.
I know from cleaning my filters that suntan lotion is by far the largest component of what is removed by the filter. My wife swims nearly every day during a 7-month swim season and wears suntan lotion that visibly gets into the water. However, the pool is only open for an hour or so each weekday (longer on weekends) therefore would not get as much benefit from the UV in sunlight so there's more chance for the suntan lotion to get caught in the filter before getting broken down. So as you say, for pools that aren't frequently used by heavy suntan lotion users, uncovered pools may not show much buildup of grease or oil in the filter.