Bogosavljević has participated in several prestigious chess tournaments and championships throughout his career. He competed in the European Chess Championship in 2013 and 2014, successfully reaching the 72nd and 75th positions respectively. In the Serbian Chess Championship, he claimed the national titles in 2012 and 2014.
The Serbian Grandmaster has also represented his country in international team events. He played for Serbia in the European Team Chess Championships in 2011, 2013, and 2015, and in the Chess Olympiads in 2014 and 2016.
In 2015, Bogosavljević won the Karpos Open in Skopje, sealing his first international tournament victory. The following year, he finished fourth in the Master Open in Zürich. He participated in the Vidmar Memorial in 2017 as well, tying for the third through seventh places.
Apart from his playing career, Bogosavljević is also a chess coach. He has been the Grandmaster in Residence for the Saint Louis Chess Club since 2020.