Vuković was born in a peasant family in Krilo near Split.
He graduated at the Classical gymnasium in Split in 1952 and at the Faculty of Law, University of Zagreb in 1956.
After a two-year military service as a reserve officer, he worked in a Secretariat of Justice and Public Administration.
[1] Vuković was elected judge of the Constitutional Court of Croatia for the first time in 1991, and later again both in 1995 and 1999.
He was one of the members of the Constituent Assembly, responsible for writing Croatian Constitution in 1990, as well as member of the Commission for war crimes (1993–1995) and State Judicial Council of Croatia (1994-1995).