Kubiak made his debut in the Polish volleyball league during the 2005/2006 season. He played for three Polish clubs before moving to foreign leagues. In the 2008/2009 season, he played for Israeli club Hapoel Kiryat Ata. His career accelerated after good performances in the B squad of the Polish national team. After that he moved to Italian Serie A2 Volley Padwa. In the 2010/2011 season, he returned to the PlusLiga, specifically to AZS Politechnika Warszawska, where he played with Zbigniew Bartman again. He wanted to leave the Warsaw club in 2011, but the club stated that their contract was still valid and filed a lawsuit. Finally, Kubiak moved to Jastrzębski Węgiel. With this club, he won a silver medal in the World Club Championship 2011. In 2012/2013 season he won a bronze medal in the Polish Championships. In 2013/2014 the club advanced to the Final Four of the Champions League in Ankara and after defeating Zenit Kazan won the bronze medal. His team beat ZAKSA Kędzierzyn-Koźle in the last matches in the fight for a medal. Jastrzębski Węgiel, including Kubiak, ended the season with a second bronze in the Polish Championships. In July 2014, a year before his contract was due to end, Kubiak requested an early termination. The club agreed, and Kubiak was able to play for the Turkish club Halkbank Ankara for the season 2014/2015. Kubiak agreed with Jastrzębski Węgiel that after the end of the contract in Turkey, he would return to the Polish club. On 29 March 2015 he won the Turkish Cup with his team Halkbank Ankara. In the final match, his team beat Arkas Izmir 3-0 and Kubiak scored 11 points. After one season in Japan with the Panasonic Panthers, he renewed his contract until the end of the 2018/2019 season.
Kubiak was first selected to represent the Polish national team by coach Andrea Anastasi in 2011. With the Polish team, he won three medals in 2011 - silver at the World Cup and two bronzes at the World League and the European Championship. He was a gold medalist at the World League 2012 in Sofia, Bulgary. On 16 August 2014 he was selected to the Polish squad at the World Championship held in Poland. On 21 September 2014 Poland won the title of World Champion 2014. On 27 October 2014 he received a state award granted by the Polish President Bronisław Komorowski, the Gold Cross of Merit for outstanding sports achievements and for promoting Poland across the world.