Kempster was born in China at Chinwangtao in May 1909. He was educated in England at Cheltenham College,[1] where he played for and captained the college cricket team.[2] From there, he progressed to the Royal Military College at Sandhurst,[1] from where he graduated into the King's Shropshire Light Infantry (KSLI) as a second lieutenant in January 1929.[3] Shortly after gaining his commission, Kempster served in the North-West Frontier Province of British India, where the Afridi Redshirt Rebellion was taking place. For his service there, he gained a medal and clasp.[1] While serving in India, he made a single appearance in first-class cricket for the Europeans cricket team against the Muslims at Lahore in the 1929–30 Lahore Tournament.[4] Batting twice in the match, he was dismissed without scoring by Mohammad Nissar, while in their second innings he was dismissed for 10 runs by Jahangir Khan.[5] Kempster was also well known in army cricket, having represented the British Army cricket team several times. Wisden described him as a "sound and attractive batsman".[2]
In the KSLI, he was promoted to lieutenant in January 1932.[6] Kempster was seconded to the Sierra Leone Battalion in April 1937,[7][8] an appointment which last until 1938.[1] It was while seconded that he was promoted to captain in December 1937.[9] By December 1938, he had returned to the KSLI and was made an adjutant in December 1938.[10] Kempster served in the Second World War with distinction. He was mentioned in dispatches in March 1945 for gallant and distinguished service in the North-West Europe campaign, at which point he held the temporary rank of brigadier;[11] he was further recognised for his service during the campaign by being made an OBE in the same month and a Companion of the Distinguished Service Order in August 1945.[12][13]
Following the end of the war, Kempster was promoted to major in January 1946.[14] He was later given the war substantive rank of lieutenant colonel, prior to his promotion to colonel in December 1950.[15] He was specially employed in January 1951, and was given the temporary rank of major general.[16] In December 1951, he was promoted to brigadier,[17] prior to his death in June 1952 at All Stretton, Shropshire.[1]