Blackman progressed through the Chelsea academy as a youth and signed a professional contract with the club in 2012. He never registered a first team appearance for the Blues; despite this, he was a member of the 2012 UEFA Champions League Final winning squad. Blackman was sent on several loan spells to Middlesbrough, Östersund, Wycombe Wanderers, Sheffield United, Leeds United, Vitesse, Bristol Rovers and Rotherham United before being released at the end of his contract in June 2021.