Born in Birmingham, Alabama, Bostock was the son of Annie Pearl Bostock and Lyman Bostock Sr. (1918–2005), a Negro leagues professional baseball star from 1938 to 1954 as a first baseman. Pearl and Bostock Sr. split when Bostock Jr. was a young child, with Pearl relocating with her son first to Gary, Indiana, in 1954. In 1958, the two again relocated, this time to Los Angeles. The younger Bostock remained estranged from his father for the remainder of his life, feeling that his father had abandoned him.