Aleksić came through the youth system at his hometown club Spartak Subotica. He made his senior debut for the club during the 2008–09 season after the club already merged with Zlatibor Voda. Aleksić made his Serbian SuperLiga debut in the 2009–10 season. He became the first choice goalkeeper after the departure of Milan Jovanić from the club, missing only one league match during the 2010–11 season. Aleksić made exactly the same amount of league games in the following two seasons.
After making his debut in early 2011, Aleksić played regularly for the Serbian national under-21 team during the qualifications for the 2013 UEFA Under-21 Championship. He made his debut for the senior side in a friendly match against Sweden on 5 June 2012.