I originally thought ALF was a metaphor for how to raise your teenager, but he is not.
ALF is the embodiment of the family’s dysfunction. He is the root of all of their squabbles and hardships. He watches too much TV; he divides the family into rival factions over certain issues/actions; he ties up all their money in groceries, impulse buys, bad investments, ponzi schemes, and more. Whenever an outsider comes to visit, the family forces ALF to hide in the kitchen or the attic, or the garage. They spend all of their time fixing what he has broken or cleaning up after him.
We learn that ALF is 229 and that he will live to be 400. When he contemplates his mortality, he envisions living with the children when they get older, he ruins their lives as well, showing that the dysfunction scars the next generation.
The only people who get to see ALF outside of the family are close family members, usually those who drop by unannounced, so that the family did not have time to properly hide their dysfunction/ALF. At some parties, friends think they catch a glimpse of ALF, but write it off as a dog or hallucination due to alcohol.
ALF has a constant desire to consume the family cat, a symbol of their love and affection. The relationship is teetering, between a balance of love and quarrel or pure dysfunction.
In the end, they work through every obstacle ALF creates and they find their lives more interesting and their bond stronger because of his presence.