What was the first TV show that had an all African American family?

The Simpsons debuted on FOX in 1989 and still airs today & will do so until May 2019 unless it gets renewed again. However, it should be noted that it's not the longest running sitcom. That belongs to the U.K.'s Last of the Summer Wine which aired for 37 years from 12 November 1973 to 29 August 2010.

Indecently it's not the longest running animated program. That belongs to Japan's Sazae-san which has been producing episodes since 1969.