Kempson was educated at Rugby School and matriculated at Trinity College, Cambridge in 1854, though there is no evidence that he took a degree.
[2] In both 1853 and 1854, he played a single first-class cricket match for the Gentlemen of England amateur team as a middle-order batsman.
Kempson joined the British Army and was a major in the 99th Regiment of Foot, where he took part in the 1860 war in China and was at the battle at the Taku Forts and the sacking of Beijing.
[2] He is recorded in the London Gazette in 1870 as transferring to the 31st Regiment of Foot with the rank of captain.
[4] He does not appear to have been related to Matthews Kempson, his near contemporary who also played first-class cricket for Cambridge University and amateur sides in the mid 1850s.