Born in Belgrade, Kljajić spent some period in Red Star Belgrade youth categories. He started his senior career with Hajduk Beograd, making 8 appearances in Serbian First League. He was declared as a man of the match 3 times with average rating 7.31. Next of Hajduk Beograd, he left to Jagnjilo, and joined Šumadija for 2009–10 season. Kljajić played 28 times, with average rating 7.31, and was declared as a man of the match 2 times. He also played for Šumadija in first half of 2010–11 season and made 13 appearances.
Kljajić signed with Metalac for second half of season, and mainly was a reserve for Dejan Bogunović. He was licensed with squad number 9, atypically for goalkeepers. He was in first 11 3 times. On the beginning of season, Kljajić changed his jersey number, and took 12. He shared his position with Živko Živković, but he was selected as a first choice in front of goal 17 times. He was also declared as the best on the field 3 times.
After Metalac's relegation from SuperLiga, Kljajić joined Rad. Coach Nikolić included him in first 11 24 times in front of his competitors Branislav Danilović, Aleksandar Jovanović and Boris Radunović. On 10 November 2013, Kljajić saved penalty kick from Dragan Mrđa. For the first half of 2013–14 season Kljajić made 14 appearances, and was declared as a man of the match 1 time. After 2014, he received the contract termination on the detriment of the club.
Partizan
On 14 February 2014, Kljajić came to Partizan and signed a four-year contract with Partizan and took the number 12 on shirt. After signing for Partizan, Kljajić was sent to Teleoptik, on loan. He collected 12 games in second half of 2013–14. Kljajić returned to Partizan at the beginning of new season, as a competitor of Milan Lukač and Živko Živković, his rival from Metalac. Coach Marko Nikolić, who also trained Kljajić in Rad included him in first 11 in Serbian Cup against Sloga Petrovac na Mlavi. He was in protocol several times, but made no league debut. He has also been on the bench four times, in UEFA Europa League, two times against Beşiktaş, one time against Tottenham Hotspur and Asteras Tripoli. During the season, he was not given a chance beside Milan Lukač and Živko Živković, mainly defended in friendly matches and several times was on the bench. On 1 November 2015, Kljajić made his debut for Partizan in SuperLiga against his former club, Metalac in 1–0 away defeat. Four days later, he made his debut for the club in Europa Leaguegroup stage against Athletic Bilbao at San Mamés. Two weeks later, Kljajić made his second cap in Europa League in a 1–2 away victory over AZ. In 2015–16 season Kljajić made 13 appearances in all competitions. After he succeeded Bojan Šaranov, Kljajić played mostly matches in the 2016–17 season as a first choice goalkeeper. At the beginning of May 2017, Kljajić earned a hand injury and missed the rest of a season. In summer 2017, Kljajić extended his contract with Partizan until 2021. He also started the next season as a first choice, but after the club signed Vladimir Stojković, Kljajić spent the rest of the same year as a back-up pick. On 30 December 2017, Kljajić moved on six-month loan deal to Platanias, without an option to purchase the contract.
