Lucky was often seen as background in the house, such as Working My Way Back to You and the Seasons 1 and 2 intro sequence. He first appeared in the pilot episode. ALF chased Lucky and tried to eat him in the pilot.
Once during a feeding frenzy, ALF is believed to have eaten Lucky, but he is later found. ALF ran off looking in the outside world, but is caught by a rescue shelter worker who doesn't know what kind of creature ALF is. ALF tricked the rescue shelter worker into releasing him so that he can grab the cat that looks like Lucky to bring home. Although it is explained that ALF did not actually grab the real Lucky, he still was not allowed to eat the cat he just took from the shelter. ALF actually tried to hypnotize the cat the same way he did to Lucky at the beginning of the episode (Episode: Looking for Lucky).
Lucky passes away in the fourth season (Episode 4x09 "Live and Let Die") and the Tanners get a new cat and name it Lucky II. Although ALF had the perfect chance to eat Lucky II while ALF never was able to do so to the real Lucky, ALF decided that Lucky II (who was a kitten) was too cute to eat.
Throughout the series, ALF was demanded not to eat Lucky. In the first episode (A.L.F.), he asked "Do you have a cat?" when he was requested if he would like anything to eat. In Looking for Lucky, he tried to hypnotize Lucky into thinking he was a bagel, but Willie caught him in the act. In Something's Wrong With Me, ALF said that one of the only ways to cure Melmacian hiccups is to drink cat juice (the hiccups are ultimately cured with raw spinach, a cure that ALF was at first reluctant to mention because it was "disgusting"). In the same episode, Kate has said to ALF, "Anything you do to this cat, I'll do to you" when ALF called Lucky "Juicy".
Lynn has twice believed that ALF actually did eat Lucky or Lucky II. Those episodes were Looking for Lucky and Live and Let Die. This proves that the Tanners indeed love Lucky very much that they are quick to accuse ALF for Lucky's disappearance.