Miloš Biković's parents divorced when he was very young, after which his father moved to Germany. Biković has since lived with his mother. His elder brother, Mihailo, is a Serbian Orthodox monk.
Biković is currently at Doctoral studies at the University of Arts in Belgrade.
Miloš Biković made his acting debut on television, in the popular RTS series The Dollars Are Coming (Serbian Cyrillic: Стижу долари). He came to further prominence with the comedy series White Ship (Bela lađa), in which he acted from 2006 to 2011. In 2008, Biković had a supporting role in the B92 drama series The Storks Will Return (Vratiće se rode). He subsequently appeared in two short films, Assignment: Ten Minutes (Zadatak: 10 minuta) in 2009, and Plus in 2010.
In 2010, Biković was cast to portray Serbian footballer Aleksandar "Tirke" Tirnanić in Montevideo, God Bless You! (Montevideo, Bog te video!). Directed by Dragan Bjelogrlić, it is the true story of the Yugoslavia national football team qualifying for the 1930 FIFA World Cup. Biković and his cast fellows had to learn to play football for the film. Montevideo, God Bless You! premiered on 20 December 2010, and became the highest-grossing film in Serbia and the region. It was also selected as the Serbian entry for the Best Foreign Language Film at the 84th Academy Awards. The film brought Biković MTV Adria Movie Award and Niš Film Festival for Best Actor, as well as a FIPRESCI Serbia Award for Best Actor nomination. It also made him a household name in Serbia, and earned him the Hello! Magazine Award for the Personality of the Year in 2011. The television series adapted from Montevideo, God Bless You!, which features some scenes and characters that do not appear in the original film, premiered on the RTS on 13 February 2012.