Born to Scottish immigrants from Falkirk, Simpson grew up in the inner-southwestern Sydney suburb of Marrickville. His father Jock was a printer and played soccer for Stenhousemuir in the Scottish League, his grandfather Harry played briefly for Stoke, and the 1910s England international player Jock Simpson was also a relative. Simpson was encouraged as a schoolboy by his two elder brothers Bill and Jack, who played in first division Sydney Grade Cricket for many years. He began his own career as a fast bowler and batsman who played in any position. He showed early leadership skills, captaining Marrickville West Primary School and later Tempe Intermediate High School. He captained 14-year-olds at the age of 12.