Bonallack was born in Chigwell, Essex. He learned the game of golf under the tutelage of head professional Bert Hodson at Chigwell and soon won the Boys Amateur Championship in 1952. A rare example of an outstanding golfer who remained an amateur in the era when professional domination of the sport became firmly entrenched, he went on to win the Amateur Championship and the English Amateur five times each and the Brabazon Trophy four times. He was a member of nine Walker Cup teams and played in the Eisenhower Trophy seven times. His best finish at the Open Championship was eleventh in 1959. He was the leading amateur at the Open in 1968 and 1971.