Hlynur started his career with Skallagrímur in the Icelandic Úrvalsdeild in 1997 but moved to Snæfell in 2002. He spent the next three seasons there, losing to Keflavík in the national finals in 2004 and 2005. In 2005 Hlynur joined Woon! Aris Leeuwarden of the Dutch Basketball League. For the season he averaged 11.8 ppg and 11.0 rpg and appeared in the Dutch all star game. He rejoined Snæfell after the season and spent the next four seasons there. Snæfell once again lost to Keflavík in the 2008 finals before finally beating them in 2010 and winning the first Icelandic championship in the history of the club. From 2010 to 2016, Hlynur played in the Swedish Basketligan for Sundsvall. In 2011 he won the Swedish championship and in 2013 and 2014 he was named the Basketligan's best defender. In August 2016 Hlynur signed with Stjarnan of the Icelandic Úrvalsdeild. On March 23, 2018, Hlynur was fouled hard by Ryan Taylor in game three of Stjarnan's first round series against ÍR. After the game he experienced concussion like symptoms and was ultimately ruled out of game four which Stjarnan eventually lost. Although Taylor received an unsportsmanlike foul, he was not ejected from the game. After the game he received a three-game suspension from the league. On April 20, 2018, Hlynur signed a two-year contract extension with Stjarnan. On May 4, 2018, Hlynur was named to the Úrvalsdeild Karla Domestic All-First Team. On 17 February 2019, Hlynur had 13 points and 14 rebounds in Stjarnan's 84–68 victory against Njarðvík in the Icelandic Cup finals. In May 2019, Hlynur was named to the Úrvalsdeild Domestic All-First Team. On 4 March 2019, Hlynur became the Úrvalsdeild all-time career assists leader, surpassing Guðmundur Bragason. Stjarnan opened the 2019–20 season with a 89–77 win against reigning champions KR in the annual Icelandic Super Cup where Hlynur posted 12 points and 7 rebounds. On 5 January 2020, he appeared in an Úrvalsdeild game for the fourth different decade. On 15 February 2020, Hlynur won the Icelandic Cup for the fourth time after Stjarnan's 75-89 win against Grindavík in the Cup final.
Hlynur was a part of the first Icelandic national basketball team to qualify for a EuroBasket tournament, in 2015. He also played at EuroBasket 2015 with Iceland, where he averaged 10.8 points and 7 rebounds per game. On September 7, 2016, Hlynur became the thirteenth player to play 100 games for the national team. Hlynur participated with Iceland at EuroBasket 2017. In February 2019, Hlynur announced that the Iceland's upcoming game against Portugal on 21 February 2019 would be his last for the national team. He retired after Iceland's 91–67 victory, where he led all players with 12 rebounds along with his 7 points. In July 2019, Hlynur agreed to return to the team for its games against Portugal and Switzerland in the EuroBasket 2021 qualification due to injuries in the squad.
Honours
Iceland
Club
Icelandic Champion: 2010
Icelandic Basketball Cup: 2008, 2010, 2019, 2020
Icelandic Super Cup: 2019
Icelandic Company Cup: 2004, 2007
Individual
Úrvalsdeild Domestic Player of the Year: 2008, 2010
Úrvalsdeild Playoffs MVP: 2010
Úrvalsdeild Defense Player of the Year: 2008, 2010, 2017