As a junior, he started his career in Szombierki Bytom, before joining Lech Poznań's academy in 2015.[4] Climbing through the ranks with junior squads and the reserve team, he was first included in the first team roster in early 2019 by Adam Nawałka.[5]
Kamiński made his Ekstraklasa debut on 20 September 2019 in a 1–1 home draw against Jagiellonia Białystok, in the starting line-up.[6] His first league goal came in a 3–3 away draw against Zagłębie Lubin on 6 June 2020. On 16 September 2020, he scored his first goal in European competitions against Hammarby IF in a 3–0 Europa League qualifying round away win. Following a successful 2021–22 campaign, which saw Lech win the championship on the club's 100th anniversary, Kaminśki was voted Ekstraklasa's Young Player of the Season.[7]
VfL Wolfsburg
On 10 January 2022, Kamiński signed a five-year contract with Bundesliga club VfL Wolfsburg, effective from 1 July 2022.[8] He debuted in a 2022–23 DFB-Pokal match against Eintracht Braunschweig, providing three assists and scoring the 2–1 winning goal.[9] On 8 March 2023, Kamiński made his Bundesliga debut in a 4–2 defeat against Bayern Munich whilst scoring his first league goal as well.[10]