Jenkinson attended Dulwich College before coming up to Pembroke College in 1901. He graduated with a first in Classics in 1904. In 1906, Jenkinson joined the staff of the Public Record Office (PRO) and worked on the records of the medieval Exchequer. In 1912, he was put in charge of the search room, and oversaw its reorganisation in response to criticisms in the first report of the Royal Commission on Public Record. During the First World War, he joined the Royal Garrison Artillery, and served in France and Belgium from 1916 to 1918. After the War he returned to the PRO and in 1938 was appointed principal Assistant Keeper. During the Second World War he was a War Office Adviser on Archives, attached to the Monuments, Fine Arts, and Archives Subcommission. After the War, until his retirement, he served as Deputy Keeper of the repository at the PRO. Away from the PRO, he lectured in London and Cambridge on many aspects of archival work. In 1922 his Manual of Archive Administration was published. It remains a key text for students of Archival Theory.