Craig was born in Chester, Cheshire, England. His mother, Carol Olivia (née Williams), was an art teacher, and his father, Timothy John Wroughton Craig, was the landlord of the pubs Ring o' Bells in Frodsham and the Boot Inn in Willington, having served as a midshipman in the Merchant Navy. Both of Craig's parents were of part Welsh descent. Craig was raised on the Wirral Penisula, and attended primary school in Frodsham, and later Holy Trinity Primary School in Hoylake. After failing his Eleven plus, Craig (along with his older sister Lea) attended Hilbre Hight School in West Kirby. Upon finishing his compulsory secondary school education at the age of 16, Craig briefly joined Calday Grange Grammar School as a sixth form student. Craig played for Hoylake RFC Rugby Union Club. He began acting in school plays at the age of six, and was introduced to serious acting by attending the Everyman Theatre in nearby Liverpool city centre with his mother. Craig moved to London when he was 16 to join the National Youth Theatre. He attended the Guildhall School of Music and Drama at the Barbican and graduated in 1991 after three years of study under Colin McCormac.
