Correct spelling, explanation: this form is correct because the noun comes from Latin familia. There is no need to double the l, as there hasn’t been any suffix added. The word family is a noun, not an adverb; -ly in this case is not a suffix but an integral part of the word. Therefore family is the correct form and familly is incorrect.

Definition of family:
noun, a group of related people, by blood, marriage, or law
Susan’s family is so lovely! Fantastic people, I like them very much.
I didn’t use to meet my family often when I was in boarding school.

Idioms and phrases with family:
runs in a family, to be a hereditary trait
Great cooking skills are something that runs in Jamie’s family.
to sell the family silver, to quickly get rid of something valuable rather than hold on to it in order to get a quick advantage
The government sold the family silver by auctioning valuable land.

Incorrect spelling, explanation: familly is a misspelling; there is only one l in the word family, just like in its origin (Latin familia). Double l is probably mistaken with adverbial suffix, in which case l doubles when adding suffix -ly. Here, however, we encounter a noun, which is one integral word: family.

Incorrect spelling

Incorrect spelling

Incorrect spelling