[5] After missing the early part of the season with his ankle injury, Kleyn recovered to be named in the starting line-up for their match against the Bulls in Pretoria,[6] but ended on the losing side as the home team won 28–12.

In the second half of 2014, Kleyn was a key player for Western Province in the 2014 Currie Cup Premier Division, starting ten matches during the season.

Kleyn established himself as a regular in the Stormers squad during the 2015 Super Rugby season, appearing in thirteen of their seventeen matches, including five starts.

[11] He once again was a key player for a Western Province team attempting to retain their Currie Cup title in the 2015 competition, playing in all twelve matches, of which he started eleven.

[12] The team finished third on the log[13] and Kleyn started their 23–18 victory over the Blue Bulls in the semi-final[14] and the final, where his side could not emulate their 2014 result, losing 24–32 to the Golden Lions.

[15] Kleyn suffered an arm injury in a warm-up match against the Jaguares prior to the 2016 Super Rugby season, which ruled him out of the start of the competition.

[37] Kleyn qualified through residency to play for Ireland on 8 August 2019,[38] and was selected in Joe Schmidt's 44–man training squad for the 2019 Rugby World Cup.

[1][2] Kleyn made his debut for South Africa when he started in their 43–12 win against Australia on 8 July 2023 in round one of the 2023 Rugby Championship,[44] and was later selected in the 33-man squad for the World Cup.