Biškupić graduated from the University of Zagreb"s Faculty of Law, and later earned a master"s degree in museology at postgraduate studies of librarian, documentation and information sciences at University of Zagreb"s Faculty of Philosophy.
He served as Minister of Culture of Croatia in the governments of three Croatian Prime Ministers: Zlatko Mateša, Ivo Sanader and Jadranka Kosor. He holds the record as the longest serving cabinet minister in a Croatian government serving for a total of 11 years and 87 days. From 1974 to 1980 he worked at a law firm based in Zagreb, and he is also notable for editing publications on visual arts published by the National and University Library.
In 1990 Biškupić joined the Croatian Democratic Union party (HDZ) and in 1992 was appointed assistant to the Minister of Culture.
From 1993 to 1995 he served as Deputy Mayor of Zagreb. From November 1995 to January 2000 he served his first term as Minister of Culture, under Prime Minister Zlatko Mateša.
After his term ended he ran in the parliamentary elections and was elected to the Croatian parliament three times at the 2000, 2003 and 2007 elections.
