A defender who can play in the middle or on the right, Cleary joined Liverpool from Dublin side Crumlin United January 2012 at the age of sixteen. Initially part of the under 16 set up, he was promoted to the under 18's in the summer of 2012 and spent two years there, making the most appearances of the squad in the 2013/14 season, when he also got three goals. In 2014/15 he was part of the under 21 squad and after being involved in the tour of Australia and the Far East, he was an unused substitute for the opening Europa League game against Bordeaux on 17th September 2015. However he did not feature in the first team picture again and was released at the end of the season, joining Birmingham City.

Cleary had a brief loan spell with Solihull Moors in the National League in late 2017, before returning to Ireland with Dundalk, where he won a league championship in 2018. He played for Dundalk in all six games when they qualified for the Europa League group stage in 2020/21 and in January 2022 joined Scottish side St Johnstone on a free transfer. He was a regular there but left in the summer, citing family reasons and he returned to Ireland, signing for Shamrock Rovers. He won the League of Ireland in his first two seasons with the club.