He became a chorister in the choir of Chichester Cathedral in 1704 and studied the organ with John Reading and Samuel Peirson.
He succeeded Peirson as Organist of Chichester Cathedral in 1720 - however he was placed under probation for a period of 13 years before he was confirmed in the office.
Kelway died in office, and left numerous compositions; his service in B minor remains a part of the choir's repertoire.
During his tenure at the cathedral, John Byfield added the second manual (choir organ).
His father was a lay vicar at Chichester Cathedral, and his brother Joseph Kelway was Organist of St Michael, Cornhill (1734-1736) and St Martin-in-the-Fields (1736-1781).