Vukšić was ordained a deacon in Mostar in 1991 and became a priest in Čitluk in 1992. He received a master's degree from the Salesian Pontifical University in 1997.
Since 2000, he served as parish priest in Slano, Posuški Gradac, Široki Brijeg and most recently in Posušje.
The previous Bishop of Kotor, Ivan Štironja, was appointed as Bishop of Poreč and Pula last year.