Kirk, a 7'0" center, played at Los Alamos High School in Los Alamos, New Mexico. He came to the University of New Mexico in 2010, averaging 4.7 points and 3.7 rebounds per game in the 2010–11 season. In the offseason, Kirk underwent back surgery for a herniated disc and subsequently redshirted the 2011–12 season. Kirk bounced back as a redshirt sophomore, averaging 12.1 points, 8.1 rebounds and 1.8 blocked shots per game. At the conclusion of the season, Kirk was named second team All-Mountain West Conference and a member of the league's All-Defensive team.
In the summer of 2013, Kirk was named to the United States' team for the 2013 Summer Universiade. He averaged 3.7 points and 4.6 rebounds as the team finished in ninth place.
In 2013–14, Kirk increased his numbers to 13.2 points, 8.7 rebounds and 2.7 blocks per game. He was named third team All-MWC and repeated as a member of the All-Defensive team.
In April 2014, Kirk declared for the NBA draft, foregoing his final year of college eligibility.
After going undrafted in the 2014 NBA draft, Kirk joined the Cleveland Cavaliers for the 2014 NBA Summer League. On August 11, 2014, he signed with the Cavaliers. During his rookie season with the Cavaliers, he had multiple assignments with the Canton Charge of the NBA Development League.
On January 5, 2015, Kirk was traded to the New York Knicks in a three-team trade that also involved the Oklahoma City Thunder. Two days later, he was waived by the Knicks.
On January 10, 2015, Kirk was acquired by the Canton Charge. In 46 games for the Charge in 2014–15, he averaged 12.5 points, 6.5 rebounds, 1.1 assists and 1.1 blocks per game.
On August 15, 2015, Kirk signed with Pistoia Basket 2000 of the Italian Serie A for the 2015–16 season. Kirk averaged 16.3 points and 7.6 rebounds per game for Pistoia, which reached the Serie A playoffs before being eliminated in the first round by Scandone Avellino.
In July 2016, Kirk played for the San Antonio Spurs in the Utah session of the 2016 NBA Summer League. He then played for the Toronto Raptors in the Las Vegas session.
On July 29, 2016, Kirk signed to play for the Guangzhou Long-Lions of the Chinese Basketball Association for the 2016-17 season. On November 15, he was released by Guangzhou.
On January 16, 2017, Kirk signed with Turkish club Anadolu Efes for the rest of the season.
On July 26, 2017, Kirk signed with Alvark Tokyo of the Japan B.League.