McKenzie attended John Curtin High School, where he performed well with both bat and ball in the school's first XI. At sixteen he was made his first grade debut for Claremont-Cottesloe as a batsman, but after unproductive performances was dropped to the second XI. The following year in 1958–59, he continued in second grade and took up fast bowling, taking 50 wickets at an average of 14.50 after his team had a pace bowling shortage. He was recalled to the First XI in 1959–60. He finished the season with 515 runs at an average of 39.46 and 49 wickets at an average of 11.21. His efforts were rewarded when the state selectors handed him a debut for the penultimate match of the Sheffield Shield season against Victoria in Melbourne. He went wicketless and scored 22 and 41. He took his first wickets in the final match against South Australia at Perth with 3/69. At season's end, his captain Ken Meuleman advised McKenzie to concentrate on his fast bowling.