He studied acting with such prominent mentors as Howard Da Silva and played character roles on and off Broadway in his mid-twenties, supplementing his income as a night watchman and shooting-gallery barker. In 1959, he appeared in his first film, The Last Mile (1959). He had a small but memorable supporting role in The Hustler (1961). In 1964 and 1965, Constantine appeared on Perry Mason, both times as the murderer. He played the role of wannabe private eye Dillard in "The Case of the Blonde Bonanza", then he played Pappy Ryan in "The Case of the Runaway Racer". In 1965, Constantine was cast as the historical John Chisum in the episode "Paid in Full", on the syndicated television anthology series Death Valley Days. He also appeared in another 1965 Death Valley Days episode "The Great Turkey War". In 1967, he appeared in the first part of "The Judgment", the two-episode conclusion of The Fugitive, starring David Janssen. Constantine played a long-suffering anti-organized-crime agent in Walt Disney's caper film The North Avenue Irregulars (1979), where he appeared alongside Edward Herrmann and Cloris Leachman. He also played an organized crime mobster who worked for Frank Nitti in the television version of The Untouchables. In 1988, he played the estranged father of one of the main characters in Friday the 13th: the Series.
