Leahy was born in Cregane, Charleville, County Limerick on the border between County Limerick and County Cork. He was one of seven brothers all of whom were sportsmen. His brother Con won medals in jumping at two Olympic Games. Another brother, Timothy, also jumped competitively. Patrick won the British high jump record in Dublin in 1898 with a jump of 6ft. 4in. (1.93m). He also competed internationally in the long jump and the hop, step and jump (triple jump).