Elliott was born David William Smith on September 21, 1960, in Milton, Ontario, Canada, the son of Pat Farrow, an office manager, and Arnold Smith, a heating and plumbing wholesale contractor. He was the second of three sons. During his teenaged years, he was part of a band, quitting Milton District High School in his final year to pursue his dream of becoming a rock star. However, at age 19, he realized this would not happen and returned to finish high school.
After graduating from high school, he attended Ryerson University in Toronto, auditioned for the Stratford Shakespearean Festival in Ontario and was accepted as a member of its Young Company.