 Gillespie was the son of Thomas Paterson Gillespie and his wife Elizabeth Hall Chalmers of Longcroft Linlithgow.He was educated at Cargilfield Preparatory School Cramond Bridge, Winchester College and New College, Oxford.He rowed for his college for three years when they were twice Head of the River and was a member of the New College eight which won the silver medal for Great Britain rowing at the 1912 Summer Olympics.