Kennedy was born John Arthur Kennedy on 17 February 1914 in Worcester, Massachusetts, the son of Helen (née Thompson) and J.T. Kennedy, a dentist. He attended South High School, Worcester and Worcester Academy. At Carnegie Institute of Technology, Pittsburgh he studied drama being graduated B.A. in 1934.
Kennedy moved to New York and, billed as John Kennedy, joined the Group Theatre. He then toured with a classical repertory company. In September 1937, he made his Broadway debut as Bushy in Maurice Evans' Richard II at the St. James Theatre. In 1939 he played Sir Richard Vernon in Evans' Henry IV, Part 1.
Kennedy got his break when he was discovered by James Cagney. His first film role was of Cagney's younger brother in City for Conquest in 1940. He was equally adept as hero or villain, and was noted for his mastery of complex, multi-faceted roles. He appeared in many Western films and police dramas.
From 1943 to 1945, Kennedy served in the military making aviation training films, both as a narrator and an actor. Many of those films today serve as an historical record of not only how aviators were trained but also how the equipment was operated.
He appeared in many notable films from the early 1940s through to the mid-1960s, including High Sierra, Champion, They Died with Their Boots On, The Glass Menagerie,Lawrence of Arabia, Peyton Place, Some Came Running, Elmer Gantry and Fantastic Voyage.
Of Kennedy's film work, he is perhaps best-remembered for his collaborations with director Anthony Mann and co-star James Stewart on Bend of the River (1952) and The Man from Laramie (1955). In both films he played sympathetic villains.
He also enjoyed a distinguished stage career over the same period, receiving a Tony Award for the role of Biff Loman in Arthur Miller's Death of a Salesman (1949). Kennedy also inaugurated three other major characters in Miller plays: Chris Keller in All My Sons (1947 ), John Proctor in The Crucible (1953), and Walter Franz in The Price (1968). In 1961 he played the title role in Becket, opposite Laurence Olivier as Henry II.
On February 5, 1959, Kennedy appeared on the episode "Make It Look Good" of CBS's Dick Powell's Zane Grey Theater. Kennedy plays Sam Carter, a former Confederate hired as a bank teller in an otherwise all-Union community by the banker Clem Doud, portrayed by Parley Baer. It is revealed that Carter, widely disliked in the town, had for a time been a prisoner of war at Elmira, New York. Carter becomes the inside partner to two brothers, played by Ed Nelson and Richard Rust, who rob the bank, but he changes his mind and does not take part in the splitting of the $30,000 loot. Carter must confront Russ Bowen, one of the brothers who had vowed to harm Carter's wife, Jenny, portrayed by Jacqueline Scott. Robert F. Simon plays Sheriff John Hanley.
With the death of his wife in 1975, failing eyesight, alcoholism, and thyroid cancer, Kennedy was reported as having lost interest in film-making. After Covert Action (1978), his next film was Signs of Life (1989).
In 1949, Kennedy won an Tony Award for best supporting actor as Biff in Arthur Miller's Death of a Salesman at the Morosco Theatre.
The New York Film Critics named him best actor for Bright Victory (1951).
His performance in Trial won him a Golden Globe Award for best supporting actor.
His portrayal of the newspaper reporter in Elmer Gantry (1960) gained him a Film Daily Award and a Limelight Award.
Kennedy and Claude Rains share the record of four losing nominations for the Academy Award for Best Supporting Actor. He also received a nomination for Best Actor, in Bright Victory (1951).
Kennedy married Mary Cheffey (1915–1975) in March 1938. They had two children: Laurie and Terence.
The last years of his life Kennedy suffered with thyroid cancer and eye disease. He died in 1990 at Branford, Connecticut of a brain tumor. He was buried in Woodlawn Cemetery, near his home at Lequille, Nova Scotia, Canada; his wife Mary is also buried there.