[1][2] Having started out at TuS SW Brauweiler, Köster completed his youth team career at TSV Bayer Dormagen, a team he would make his senior handball debut with in 2018, playing in the 2.

[3] In 2020, the German switched to league rivals VfL Gummersbach, signing a contract until 2024.

[5] He became captain of the team in 2022, and signed on to play for Gummersbach until 2026 following a tenth-placed finish in the 2022–23 Bundesliga campaign.

[6][7] Köster made his international debut on the German national team on 21 September 2021.

[8] Afterwards, Köster took part in the 2022 European Championship, where he played in all seven matches and became Germany's second-highest scorer with 18 goals.