Growing up in Kilburn, London, he attended high school at City of London School, where he was in the same class as Harry Potter actor Daniel Radcliffe. At the age of seven, Crawford was diagnosed with alopecia universalis, a rare, autoimmune disease causing the total loss of all hair on the body.
After moving to the United States in 2005, he attended St. Augustine College Preparatory School while he resided in Longport, New Jersey. Although he intended to pursue basketball, he couldn't play as a sophomore because of an international transfer rule and decided to join the American football team with no previous experience.
He found success as a defensive end and tight end in his last year, being rated as the fifteenth defensive end in the nation (Scout.com), registering 12 touchdown passes and receiving All-area honors. He also practiced basketball and football.
Crawford accepted a scholarship from Pennsylvania State University. As a true freshman he played both defensive tackle and tight end, recording 4 tackles and 2 passes defensed. The next year, he replaced Aaron Maybin and started all games at defensive end, ranking ninth in the Big Ten Conference in tackles for loss (15) and tenth in sacks (5.5).
As a junior started 7 games and recorded 14 tackles (5 for loss) and 2 sacks. In his last year he started all games, posting 40 tackles (7 for loss), 6.5 sacks,6 passes defensed and one fumble recovery.
Crawford was selected by the Oakland Raiders in the fifth round (158th overall) of the 2012 NFL Draft. As a rookie, he was a backup player, appearing in 4 games and registering 5 tackles.
The next year, he played all along the defensive line, appearing in 15 games, while recording 12 tackles (one for loss) and 2 quarterback pressures. He was waived on August 30, 2014.
On September 2, 2014, Crawford was signed as a free agent by the Dallas Cowboys, who had scouted him during joint training camp practices with the Oakland Raiders. He missed 3 games with a calf injury before returning to play in London against the Jacksonville Jaguars, registering 2 tackles, one sack, one quarterback pressure and forced a fumble. He suffered a right broken thumb against the New York Giants and was placed on the injured reserve list on 26 November.
In 2015, he played in every game (one start) at both defensive tackle and defensive end, finishing with 18 tackles (3 for loss), 4 sacks, 10 quarterback pressures and one pass defensed.
Crawford re-signed with the Cowboys on a one-year, $1.1 million contract on April 4, 2016. He made a career-high 10 starts, mostly at left defensive end, posting 27 tackles, 3.5 sacks and 12 quarterback pressures (fifth on the team). During the games he was used at multiple positions along the defensive line.
On March 9, 2017, Crawford signed a three-year contract with the Atlanta Falcons.