The original Charles Addams cartoons were just single-panel gags in The New Yorker, so they didn't go into detail. The characters didn't even have names until Addams was asked to come up with them for the TV series. There was something monstrous about them, but it wasn't specified exactly what. There were hints of the supernatural, like the two cartoons that were combined as the inspiration for Thing: a gag of a record player whose automatic record-changing arms were disembodied humanlike hands, and a sign on the fence outside the house saying "Beware of the Thing."
In adaptations, Grandmama (Morticia's mother) is explicitly a witch, and in the original sitcom, Morticia was implied to be one as well, occasionally showing unearthly powers like her "smoking" trick (where she asks "mind if I smoke?" and then smoke spontaneously emerges from her body). Uncle Fester's light-bulb trick in the screen versions suggests there's something artificial about him, except he's treated as a member of the biological family (Morticia's uncle in the sitcom, Gomez's brother in later versions). Gomez seems to be just a rich eccentric, except that Cousin Itt is a member of his family. The Thing is clearly supernatural -- arguably more so in the sitcom, where he emerges from boxes that evidently link to some extradimensional space, than in later versions where he's just a severed hand running around on his fingertips. Lurch is often implied to be a Frankenstein's Monster, though in the sitcom he had a mother.
I've never been huge into the Addams Family, but this does look pretty interesting.
