[7] He played for various youth teams at FSV Gevelsberg, SSV Hagen and VfL Bochum.

By the end of June, it was revealed that Klostermann had yet to sign a professional contract, leading to a start of negotiating.

[17] In August 2014 it was announced that Klostermann and VfL Bochum had failed to reach an agreement regarding the extension of his contract.

[18] Subsequently, he did not play another game for Bochum and on 22 August 2014 it was announced that he would transfer to RB Leipzig.

[21] He made his RB Leipzig debut against Erzgebirge Aue in the second round of DFB-Pokal on 29 October 2014 but scored an own–goal and helped the side win 3–1 later in the game.

[22] It wasn't until on 12 December 2014 when he made his league debut for the club, where he came on as a late substitute, in a 1–0 win over Greuther Fürth.

[30] Despite missing out three matches, due to injuries later in the 2015–16 season,[31] Klostermann went on to make a total of 31 appearances and scoring once in all competitions.

Ahead of the 2016–17 season, Klosterman was linked with a move away from RB Leipzig, as Premier League duo Tottenham Hotspur and Arsenal showed interest.

[36] The 2017–18 season saw Klostermann regaining his first team place at RB Leipzig, playing in the right–back position.

[49] Klostermann helped the club reach their first DFB-Pokal Final after beating Hamburger SV 3–1.

[54] He made his Germany U17 debut, where he played 40 minutes after coming on as a substitute, in a 3–1 win over Netherlands U17 on 10 February 2013.

[63] Almost a year absent, Klostermann was called up for the Germany U21 squad again and played 45 minutes, in a 2–1 loss against Hungary on 1 September 2017[64] Two months later, he scored two goals in two matches against Azerbaijan U21 and Israel U21.

[65] On 27 March 2018, Klostermann captained the U21 side for the first time, starting the whole game, in 0–0 draw against Kosovo U21.

[70] Despite this, Klostermann were among six Germany U21 players to be named UEFA European Under-21 Championship Team of the Tournament.

[72] Klostermann played his first match of the tournament on 4 August 2016, starting the whole game, in a 2–2 draw against Mexico U23.

[76] After the tournament, Klostermann revealed that he auctioned his silver medal, alongside Alexandra Popp.

[85] Off the field, Klostermann maintains a private persona, preferring to focus on his career rather than seeking media attention.