Kupkovič was born in Bratislava, and studied violin and conducting there, first at the conservatory, then at the Academy of Performing Arts. He played violin in the Slovak Philharmonic Orchestra from 1960 to 1965, and then began to write music for television and film to make a living. At the same time, he was writing more experimental music for concerts. In 1969 he won a music scholarship to West Berlin, and emigrated there the following year. In 1971, he conducted the premiere of Karlheinz Stockhausen's Mixtur in Cologne, a piece dedicated to Kupkovič himself. In the same year, he began to teach music theory at the Hochschule für Musik und Theater Hannover, and lives in Hanover today. During the 1970s he was associated with the Cologne School (Kapko-Foretić 1980, 50).