Burkill was born in Holt, Norfolk, and educated at St Paul's School and Trinity College, Cambridge, where he won the Smith's Prize.
[2] He became a research fellow at Trinity in 1922, and two years later was appointed Professor of Pure Mathematics at Liverpool University.
[2] In 1929, he returned to Cambridge to take up a position as Reader in Mathematical Analysis, as a fellow not of Trinity but of Peterhouse.
In 1928 he married Margareta Braun, who was born in Germany but educated at Newnham College, Cambridge.
[5] After Margareta's death in 1984 Burkill lived in Sheffield, where his adopted son Harry was based, and died there in 1993.