Miljković was born on January 8, 1962, in Kragujevac, Yugoslavia/Serbia. He started playing chess at a young age and gradually garnered international recognition. His play style is considered to be technical and tenacious, and his favorite opening moves are the Sicilian Defence and the French Defence.

The grandmaster has achieved a number of significant placings in international tournaments. These include winning the Yugoslav Championship in 1988, earning fifth place in the World Junior Chess Championship in 1982, and competing in the Chess Olympiad representing Yugoslavia in 1992.