Alexander was born at Britwell Salome, Oxfordshire, England. Alexander was a forceful bat and a fast bowler, and also acted as the manager of the Australian teams that toured England under Billy Murdoch in 1880 and 1884. He also managed the England tour of 1882-83 under Ivo Bligh that went to Australia to retrieve The Ashes.
In 1880, in addition to his management duties, Alexander was one of the leading bowlers of the team that visited other colonies as well as England. On the tour as a whole, he took 109 wickets at an average of nine runs each. He played in the first Test match on English soil at The Oval, took two wickets and with his captain added 52 runs for the ninth wicket, which helped avoid an innings defeat. He played a second Test match four years later at the Adelaide Oval, but did not take a wicket.