Weston accepted a football scholarship from the University of Florida, and played for head coach Galen Hall's Florida Gators football team from 1985 to 1988. As a freshman, he was a backup and had 2 sacks in the third game of the season against Mississippi State University. The next year he was named the starter at right defensive tackle and posted 96 tackles, including a season-high 15 tackles against the University of Georgia. As a junior in 1987, he was a first-team All-SEC selection and an honorable-mention All-American, after finishing the year with 54 tackles, 14 tackles for loss and 6 sacks, although he played the final four games with a broken jaw he suffered in a nightclub fight. He was a part of a defense that was third in the nation in pass defense, fourth in total defense and recorded back-back shutouts for the first time in school history since 1956. As a senior, Weston was part of a dominant defensive line, along with Trace Armstrong and Jeff Roth. He suffered injuries in the preseason that kept him from starting 2 of the season's first 5 games. He returned against Mississippi State University, tallying 4 tackles, 2 passes defensed and his only career interception. Against Memphis State University, he had a season-high 10 tackles, one sack and one forced fumble. He finished the year with 49 tackles, 15 tackles for loss, 10 sacks, 5 passes defensed, one forced fumble, was a second-team All-SEC selection and an honorable-mention All-American.
Weston was selected by the Dallas Cowboys in the third round of the 1989NFL Draft. After having a disappointing training camp, he was waived on August 29 and replaced with defensive tackle Dean Hamel.
In 1989, Weston was claimed by the Tampa Bay Buccaneers and eventually released on September 4. He was signed to the team's practice squad on September 6. He was promoted to the active roster during the regular season and played in twelve games, with two starts, at defensive end. He had two quarterback sacks and a fumble recovery.
Weston mother died in a car accident when he was a junior in college. He donated one of his kidneys to childhood friend Tommy Duhart, who had battled severe diabetes. He later returned to Gainesville and completed his bachelor's degree in health and human performance, graduating from the University of Florida in 2010.
