He attended Palm Beach State College[2] before graduating from the University of Georgia with a master's degree in theater.

[4][5][6] Markham played the dual roles of Luke and Ken Carpenter in the 1967–1968 ABC sitcom The Second Hundred Years,[3] and portrayed Harry Kellem in the original Hawaii Five-O.

On 9 November 1975, in the third-season episode "The Bionic Criminal", Markham reprised the role – although with the character's name changed to "Barney Hiller".

Markham appeared as Blanche Devereaux's gay brother, Clayton Hollingsworth, on the NBC sitcom The Golden Girls, in two episodes.

[citation needed] One of his earliest stage experiences was at the Oregon Shakespeare Festival in the summer of 1961, where he played Horatio in Hamlet.