He studied for the Priesthood at Oscott College where he was ordained on 29 June 1937 by Archbishop Thomas Leighton Williams.

Returning to the Salford Diocese in 1940, Fr Burke was appointed as Professor at St Bede's College, becoming Prefect of Studies in 1950 and Vice-Rector in 1953.

At St Bede's he taught Ian Kershaw who later said Burke was "a marvellous teacher and I owe him a lot".

[1] In 1966 following the retirement of Monsignor Thomas Duggan through ill health, Burke was appointed Rector of the college but only remained in the post for a year before being appointed Auxiliary Bishop of Salford by Pope Paul VI in 1967 to assist Bishop Thomas Holland.

Bishop Burke took up residence at Nazareth House, Prestwich, and then later at the Little Sisters of the Poor in Longsight, where he died on 13 October 1999, aged 86.