Cleary grew up in Drimnagh, Dublin and played for local side Crumlin United at schoolboy level, the club that produced several international players including Robbie Keane.
[3] Cleary made his Liverpool first team debut on 2 August 2015 in a pre-season friendly against Swindon Town, coming off the bench in the 57th minute replacing Trent Alexander-Arnold at right back in a 2–1 win at the County Ground.
[16] On 19 January 2018, after returning from his loan at Solihull Moors, Birmingham announced that Cleary had left the club by mutual consent,[17] later admitting that he had fallen out of love with the game at that time.
[18] On 26 July 2017, Cleary sealed a loan move to National League side Solihull Moors until 2 January 2018, alongside Birmingham teammate Connal Trueman.
[23] Cleary won the first trophy of his career on 5 October 2018, as his side clinched the 2018 League of Ireland Premier Division with a 1–1 draw against St Patrick's Athletic.
[30] On 1 October 2020, Cleary scored in a 3–1 win against KÍ of the Faroe Islands at the Aviva Stadium to seal qualification to the UEFA Europa League Group Stages.
[21] On 6 December 2020, Cleary played the full 120 minutes as Dundalk beat Shamrock Rovers 4–2 after extra time to win the 2020 FAI Cup Final.