Originally from Reading, Cooper joined QPR at the age of 10, coming through their Academy system.

Cooper signed a professional deal at Loftus Road in 2021, and had loans with Welling United, Bath City and Stratford before making his Hoops debut towards the end of the 2021/22 season. 

In 2023/24, Murphy joined Swindon on a long-term loan, in Sky Bet League Two, and then was borrowed by Stevenage the following season, in League One, playing 40 games. 

During 2025/26, Cooper spent the first half of the season at Barnsley, making his debut against Argyle on the opening day, and then headed to Sheffield Wednesday, playing eight times in the Championship.