Kennedy was born in Brighton, England, on December 28, 1956. He studied piano in his early childhood with his cellist mother. Later on, as a student at the Yehudi Menuhin Music School, he turned to the violin. In his teens he enrolled at Juilliard where he studied under Dorothy DeLay.
At 16 he debuted at Carnegie Hall with jazz violinist Stephane Grapelli. Around this time Kennedy began delving heavily into jazz, regularly performing with jazz musicians in New York.
By the early '80s Kennedy had garnered international recognition, and in 1984 made his recording debut with a critically acclaimed performance of the Elgar Violin Concerto on EMI. But he withdrew from public performance from 1992-1997.
Kennedy was honored at the 1997 BRIT awards with an Outstanding Contribution to British Music award. In 1999 Sony issued a recording entitled The Kennedy Experience, which contained improvisations based on music of rock guitarist Jimi Hendrix
Kennedy's first bona fide jazz recording, The Blue Note Sessions, was issued with great success in 2006 on Blue Note Records. Two years later Kennedy made his first appearance at the Proms in 21 years, with a performance of the Elgar Violin Concerto. Among Kennedy's recordings is the 2008 CD pairing the Beethoven and Mozart fourth violin concertos on EMI.