Mitchell was born in Dundee, Scotland, on 22 June 1921. His father was a blacksmith. Mitchell went to school at Morgan Academy, Dundee. He played football at school and was invited to sign for North End Junior Football Club in Dundee. He left Morgan Academy in 1938 after receiving a scholarships through the school to do a mathematics degree in the University College, Dundee. He graduated with First Class Honours in 1942, and was called up and sent to the wartime Ministry of Aircraft Production in London, where he remained until after the end of the war. While he was in London, he continued to play football during the war, turning out a few times for Chelsea. After the war he played for a number of Scottish clubs including St. Johnstone, East Fife, Brechin City before ending his playing career at Berwick Rangers in 1955.