Kirke was born in Lambeth, South London, the son of Olive May and Vivian Percy Kirke. He spent his early years living in the countryside of Wales. Leaving school at 17, he returned to London and set about finding a drumming job in the booming blues scene. After a fruitless 22 months he was resigned to returning to the country when he met Paul Kossoff who was playing in a band called Black Cat Bones. Kirke was offered the drumming position in the band for which he played for six months.
Kirke and Paul Kossoff left the band and with Paul Rodgers and Andy Fraser formed Free. In the four years Free were together they had an enormous influence on many bands on both sides of the Atlantic. Their biggest hit, "All Right Now", was a number one in more than 20 territories and was acknowledged by ASCAP (American Society of Composers, Authors, and Publishers) in 1990 as having received more than 1,000,000 radio plays in the US by late 1989. In 2000 an award was given to Paul Rodgers by the British Music Industry when "All Right Now" passed 2,000,000 radio plays in the UK.
By April 1972 Free had reformed with Andy Fraser and Paul Rodgers making peace, and Kossoff appeared to pull it together if only briefly. Kossoff resumed taking drugs during the US tour to support the last album by the original quartet Free at Last. On the eve of their Japanese tour Fraser fought with Rodgers and once again left the band, to be replaced by Tetsu Yamauchi. Rodgers and Kirke elected to fill in the band bringing John "Rabbit" Bundrick on board as a member of Free for the tour and the last Free album Heartbreaker. After the disbanding of Free in 1973, Kirke and Rodgers again teamed up to form Bad Company. They were joined by guitarist Mick Ralphs (Mott the Hoople) and bassist Boz Burrell (King Crimson).
After Bad Company disbanded in 1982, Kirke was briefly involved with another band, Wildlife. Wildlife toured in support of the Michael Schenker Band around this time in the UK; their album was produced by former Bad Company bandmate Ralphs, and featured Kirke playing saxophone on one track, "Charity". Despite being signed to Led Zeppelin's label, Swan Song, and Kirke's high-profile involvement, the album failed to sell. Wildlife's core members, brothers Chris and Steve Overland went onto greater success with the band FM. Kirke returned to Bad Company when the band reformed in 1986. As well as touring with Ringo Starr's All Star Band on three occasions, he is an accomplished songwriter, releasing Seven Rays of Hope in 2005. He toured with Bad Company in 2009. He has been playing with New York City rock band Zeta Vang as a side project.