Born in New York City, Bissell was the son of prominent surgeon Dr. J. Dougal Bissell. He trained with the Carolina Playmakers, a theatrical organization associated with the University of North Carolina at Chapel Hill. He had a number of roles in Broadway theatre, including the Air Force show Winged Victory, when he was an Airman. In a career that began with the film Holy Matrimony (1943), Bissell appeared in hundreds of films and television series episodes as a prominent character actor. Regularly cast in low-budget science fiction and horror films, these include a mad scientist in the film I Was a Teenage Werewolf (1957) and as Professor Frankenstein in I Was a Teenage Frankenstein (also 1957). He also played the doctor who treats Kevin McCarthy’s character in Invasion of the Body Snatchers (1956) and in the original version of Creature from the Black Lagoon (1954). Bissell appeared as a guest star in many television drama series between the early 1950s and the mid-1970s, with more sporadic appearances after that. He guest-starred on Rod Cameron’s syndicated City Detective (1955) and around the same time in a couple of episodes of The Lone Ranger. He also appeared on other syndicated series, including Sheriff of Cochise, Whirlybirds, and The Brothers Brannagan. He was cast too in the religion series, Crossroads and Going My Way, and in the NBC education drama series, Mr. Novak. He also appeared as Lt. General Heywood Kirk in 30 episodes of ABC’s sci-fi series “Time Tunnel” during the 1966-67 season. Bissell appeared in the western series, Cheyenne (“The Broken Pledge”, 1957) playing General George Armstrong Custer. The episode focuses on the betrayal of the Sioux. Also in the cast William Fawcett, Gary Vinson, and John Dehner.

Bissell played murderer Larry Sands on CBS’s Perry Mason (“The Case of the Crooked Candle”, 1957). He made three other Perry Mason appearances, including the role of Max Pompey in “The Case of the Lavender Lipstick” (1960), and Laurence Barlow in “The Case of the Nautical Knot” (1964). He appeared in Fireside Theater (“The Man on the 35th Floor”, 1957) with fellow guest stars Macdonald Carey and Phyllis Avery, and as brutal television critic Otis Elwell in Playhouse 90 (“The Comedian 1957), with Mickey Rooney and Mel Tormé. He had a role in the western series, The Californians as Dr. Haley (“Pipeline” , 1958). In 1959, Bissell appeared on the NBC science fiction series, The Man and the Challenge. He guest starred on the CBS western series, The Texan, starring Rory Calhoun, and Johnny Ringo, with Don Durant in the title role. He played different roles in multiple episodes of the ABC series The Rifleman. In 1960, Bissell portrayed Judge Seward in the episodes “Brother’s Keeper” and “Judge Seward’s Secret” on the ABC/WB crime drama, The Roaring 20s. In the story line, the judge tries a case against a mobster named Rossi, played by Herman Rudin, despite threats on his life and his family. In 1960, he was cast on an episode of another ABC/WB crime drama, Bourbon Street Beat, and on the CBS anthology series, The DuPont Show with June Allyson. From 1959 to 1961, Bissell was a regular for the third and fourth seasons of the television series Bachelor Father, costarring John Forsythe, Noreen Corcoran, and Sammee Tong. In 1961, he guest starred in the episode “A Fool for a Client” on James Whitmore’s The Law and Mr. Jones legal drama on ABC. He also appeared in the ABC adventure series, Straightaway. He was cast three times on the long-running NBC western series, The Virginian and once on the short-lived ABC/WB The Dakotas. He portrayed “Clement Samuels” in “Kiz”, an episode of the Western comedy series Maverick starring Roger Moore and Kathleen Crowley.

Bissell’s most prominent television role came when he co-starred as General Heywood Kirk in the 1966-1967 science-fiction television series The Time Tunnel. He often played silver-haired figures of authority, here as in many other roles (as described by Allmovie), “instantly establishing his standard screen characterization of fussy officiousness”, leavened in many instances with a military bearing. Other examples of such authoritative roles as military or police officials, include appearances in The Caine Mutiny, The Manchurian Candidate, The Outer Limits (1963), Hogan’s Heroes (1966), and The Man from U.N.C.L.E. (1966). Bissell appeared in the classic episode “The Trouble With Tribbles” of Star Trek, footage of which was re-used in Star Trek: Deep Space Nine’s “Trials and Tribble-ations”. In 1978 and 1980, Bissell appeared in episodes of The Incredible Hulk, first in the second season episode “Kindred Spirits”, and next (and lastly) in the second part of the fourth season two-parter “Prometheus”. He played a different professor in both episodes. Bissell portrayed the undertaker (who sees every man, no matter his race, as “just another future customer”) in the film The Magnificent Seven (1960). In 1960, Bissell had appeared in George Pal’s production of The Time Machine, as Walter Kemp, one of the Time Traveller’s dining friends. He also appeared in a 1978 TV movie adapting the H.G. Wells novel for a more modern setting. Thirty-three years later, in 1993 the documentary film Time Machine: The Journey Back (which featured Bissell, Rod Taylor and Alan Young), he recreated his 1960 role as Walter in the opening sequence. It was Bissell’s last acting performance.

Bissell received a life career award from the Academy of Science Fiction, Fantasy & Horror Films in 1994. He also served for many years on the board of directors of the Screen Actors Guild, and represented the actors’ branch of the Academy of Motion Picture Arts and Sciences board of governors. Bissell died in 1996 in Woodland Hills, California from the effects of Parkinson’s disease. He was interred in the Westwood Village Memorial Park Cemetery in Los Angeles.