Mitchell was born in Columbus, North Carolina, to an African-American family, and moved as a young child with his parents to Harlem, New York City.
As a high school student, he began performing and writing theatrical sketches, and joined the Rose McClendon Players.
He met performers such as Ethel Waters and George Wiltshire, and encountered racial discrimination at first hand in his everyday life.
He became a graduate student at Columbia University between 1947 and 1951, studying playwriting, while also working as an investigator for the Department of Welfare.
His 1957 play A Land Beyond the River was a fictionalised adaptation of the life of schoolteacher and pastor Joseph DeLaine, whose lawsuit helped end segregation in public schools in the United States.