In spite of their domestication, house cats are still predators. And all predators -- from polar bears to ladybugs -- clean themselves after they eat. Any cat that neglects her washing ritual needs to see her veterinarian, because a lack of grooming is abnormal in this fastidious animal.
Cats will groom at any time of day. As hunters, they must keep themselves clean to avoid any odor that would alert prey prematurely to their presence. This is more important for those that are ambush predators, those that lie in wait for their prey (leopards, cougars, house cats), than for those that run their prey down (cheetahs). Lions, which use a combination of both tactics, will frequently have a period of socialization and mutual grooming before an evening hunt. Domestic cats, like lions, will groom each other with great zeal. A cat's grooming before eating is a natural instinct.
Cats will sometimes pause while eating to lick their lips and whiskers or lick the food rather than bite into it. A cat eating a kill, whether a house cat with a mouse or a lion with a zebra, will use its very rough tongue to collect blood and stray bits of meat from its face and paws as it feeds. Cats can actually use this uniquely modified organ to strip bones of meat their teeth cannot access.
After eating is when grooming is likely to be most rigorous. Successful hunters must rid themselves of all traces of blood to keep other and more powerful predators from homing in on the scent and turning the hunter into the hunted. At this time, if at no other, tongue-washing is likely to start at the lips, then move to a forepaw, with which the cat will clean the face and ears, then the other forepaw and the side of the face, and so on down the chest, forelegs, body, hind legs and belly, all the way to the tip of the tail (not forsaking the intimate personal zone, which requires minute attention in astonishing postures).
Failure to Groom
Stray or abandoned cats might be unkempt simply because they can't groom -- scraping a living takes all their time and energy. Outdoor pet cats and barn cats generally are better-groomed because they have easier access to food from humans or at least better hunting territory, and therefore more time to groom.
An indoor cat that shows self-neglect -- such as a matted or greasy-feeling coat, dirty feet and an unpleasant smell about her -- may be in serious trouble. She may be sick or have dental problems. Elderly cats may have arthritic joints that make grooming certain areas painful or impossible. Fat cats may not be able to reach parts of their bodies to clean them. If your cat's face, chest and paws aren't sparkling clean half an hour after a meal, watch her carefully to see if she has a problem and take her to the vet if you think she does.