Kleinen joined the MVV youth academy from his first club RKSV Leonidas-W in 2011.[3] In the 2019–20 season, he featured as an unused substitute for the first team on several occasions. He made his professional debut on 30 August 2020, replacing Tim Zeegers in the 79th minute of a 0–0 away draw in the Eerste Divisie against Almere City.[4]
On 13 February 2021, Kleinen made his first ever start for MVV in a 1–0 away defeat to De Graafschap.[5] Four days later, he signed his first professional deal with MVV; a two-year contract.[3]
Kleinen scored his first goal at senior level on 30 September 2022, starting MVV's eventual 3–2 second-half comeback win over Willem II by curling home a shot into the right-hand corner.[6][7] On 21 April 2023, he extended his contract with MVV until 2025 with an option for an additional year.[8][9]