[1] His parents and older brother left Montenegro during the breakup of Yugoslavia in the early 1990s and settled in Sydney before Vukic was born.

[2] Vukic began playing tennis at the age of 6 and attended Normanhurst Boys High School throughout his upbringing.

Following a quarter-final appearance at 2019 Internazionali di Tennis Città dell'Aquila, Vukic reached a career high singles ranking of No.

[5] In September, Vukic qualified for the 2020 French Open main draw, where he made his Grand Slam singles debut.

Vukic commenced the 2021 season at the 2021 Great Ocean Road Open, where he defeated Yen-Hsun Lu for his first ATP main draw win.

[8] At the 2021 Australian Open, Vukic entered into the main draw as a wildcard and lost to 19th seed Karen Khachanov in the first round.

[10] In October at the 2021 BNP Paribas Open he recorded his first main draw win at a Masters 1000 level as a qualifier defeating Pablo Andújar.

At the Adelaide International 2, Vukic achieved his first top 50 win against Alexander Bublik[13] and reached his first ATP quarterfinal, before losing to Thanasi Kokkinakis.

At the 2023 Delray Beach Open he entered the main draw as lucky loser and won in the first round against qualifier Christopher Eubanks.

He won his third professional title and the biggest of his career, a second Challenger at the 2023 Busan Open in South Korea, defeating top seed Max Purcell by retirement.

He reached the third round of a Masters 1000 for the first time at the 2023 Canadian Open as a lucky loser, defeating Borna Ćorić and Sebastian Korda.

[20][21] At the beginning of the European clay season at the 2024 Grand Prix Hassan II, he reached the quarterfinals defeating previous year runner-up Alexandre Müller, and then second seed Sebastian Ofner in a match with two tiebreaks lasting close to three hours.

Next, also as a lucky loser, he entered the main draw of the 2024 Eastbourne International and reached his second ATP semifinal with wins over Fabian Marozsan, third seed Alexander Bublik and Yoshihito Nishioka (who beat him in the qualifications).