On 21 March 2011, Cleaver made his first-class debut for Central Districts in a rain-affected draw, where only 8.2 overs were possible in the match.[3] He made his List A debut the following season.
In June 2018, he was awarded a contract with Central Districts for the 2018–19 season.[4] In February 2020, in the 2019–20 Plunket Shield season, Cleaver scored a double century in the first innings and made nine dismissals.[5]
In March 2022, Cleaver was named captain of the Central Districts cricket team for their Plunket Shield match against Northern Districts in Napier, Hawke's Bay at McLean Park.[6]
In June 2022, Cleaver was added to New Zealand's Test squad, for their third and final match against England,[10] but did not play.
Later that same month, he was named in New Zealand's One Day International (ODI) and T20I squads for their tours of Ireland, Scotland and the Netherlands.[11] Cleaver made his T20I debut on 18 July 2022, for New Zealand against Ireland,[12] and in the second match of the series he hit a match-winning 78 not out.[13]
Cleaver made his ODI debut on 31 July 2022, for New Zealand against Scotland.[14]