Cleary was born in Tokyo, Japan, but moved to the United States where he first learned football. He graduated from St. Viator High School in Arlington Heights, IL and was called "The Asian Sensation" by students. He played as an offensive tackle and defensive end. As a senior he received honorable-mention All-State honors. He also was a two-time All-East Suburban Catholic Conference selection.
Cleary accepted a football scholarship from Boston College. He established himself as a starter at right tackle for the last 5 games of his sophomore season, after Rich Lapham was injured. He was moved to left tackle as a senior.
On April 13, 2013, Cleary was signed as an undrafted free agent by the Indianapolis Colts. After competing for a roster spot during their training camp he was waived/injured (undisclosed) on September 5.
On October 16, 2013, Cleary was signed by the Tampa Bay Buccaneers to their practice squad. He was released on June 17, 2014.
On June 18, 2014, he was claimed off waivers by the Oakland Raiders. He was released on August 24.
On September 16, 2014, he was added to the Cincinnati Bengals practice squad.
On January 13, 2015, Cleary signed a reserve/future contract with the New York Giants. He was waived on September 5. On September 6, he was signed to the Giants' practice squad. On November 16, he was elevated to the active roster. On September 6, 2016, he was waived/injured (finger).
On September 28, 2016, Cleary was signed by the Dallas Cowboys to be the reserve swing tackle and replace an injured Chaz Green. He played only on special teams until Week 16 against the Detroit Lions, where he replaced an injured Tyron Smith at left tackle in the third quarter. He started at left tackle in the final game of the season against the Philadelphia Eagles in place of an injured Smith.
In 2017, he was passed on the depth chart by Byron Bell during training camp and was waived on September 2.
On September 12, 2017, Cleary was signed by the Detroit Lions.