Clayborn was born in St. Louis, Missouri. At birth, he suffered from a condition known as Erb's Palsy, something that is caused during birth by an injury to the nerves surrounding a child's shoulder. Because his head and neck were pulled to the side as his shoulders passed through the birth canal, he suffered nerve damage resulting in the loss of some movement and weakness in his right arm. Clayborn underwent physical therapy throughout his young life and eventually overcame the limitations caused by the disorder.
When Adrian was 10, his older brother, Anthony, was shot and killed in northern St. Louis.
Clayborn attended the University of Iowa, and played for the Iowa Hawkeyes football team from 2006 to 2010. He took a redshirt year in 2006, his first season at Iowa. In 2007, Clayborn became a second-string defensive lineman and saw action in several games on special teams. In 2008, Clayborn saw substantial playing time, recording 8 tackles for loss and 50 total tackles. In 2009, Clayborn had a breakout year with 20 tackles for loss. In a dominant effort against Georgia Tech, Clayborn was named Orange Bowl MVP.
Heading into 2010, Clayborn was listed on the preseason watch list for several awards.ESPN, Playboy Magazine, Sporting News, College Football Insiders, Lidy's and Phil Steele all selected Clayborn for their pre-season first team All American lists.
On October 2, 2010, Clayborn had his first break-out game of 2010. He had three tackles for loss against Penn State despite being triple-teamed for parts of the game. His play earned him Big Ten Defensive Player of the Week. On November 10, 2010, Clayborn was named one of four finalists for the Rotary Lombardi Award.
While preparing for the 2010 Insight Bowl, Iowa head coach Kirk Ferentz announced that Clayborn was one of three Hawkeyes invited to attend the Senior Bowl, college football's pre-draft event featuring seniors with NFL prospects.
The Tampa Bay Buccaneers became the first team to sign their first-round pick by signing Clayborn, according to ESPN. Clayborn recorded his first sack in week 3 of the 2011 season against the Atlanta Falcons. His sack on Matt Ryan also caused a fumble recovered by teammate Michael Bennett. In 2012, Clayborn was placed on injured reserve after injuring his knee.