Liverpool signed Kelleher from Ringmahon in the summer of 2015. He featured regularly in the under-18s and under-23s while also gaining experience in the UEFA Youth League. His progress was rewarded with performances for the first team in the International Champions Cup in the summer of 2018 as well as featuring in a friendly victory over Blackburn Rovers. The goalkeeper was on the bench when Liverpool faced Wolves in the third round of the FA Cup on 7 January 2019 and was named third-choice keeper for the Champions League knockout stages. Changes to regulations over substitutes meant he was on the bench for the final and took part in the on-pitch celebrations after the triumph in Madrid. Kelleher made his long-awaited first-team debut when he delivered an impressive performance against Milton Keynes Dons in the Carabao Cup and he also played against Arsenal and Aston Villa in the competition, as well as against Shrewsbury in an FA Cup fourth round replay at Anfield.

Kelleher challenged Adrian for the role of back up keeper in 2020/21, appearing twice each in the Premier League and Champions League. At the end of the season he made his full international debut, coming on as a second half substitute in Ireland's friendly with Hungary. During the close season he signed a contract extension, keeping him at Liverpool until 2026. In 2021/22 Kelleher was rewarded for his performances in the Carabao Cup by being named as keeper for the final against Chelsea. He became the match winning hero at the end of a marathon penalty shoot out in which all twenty outfield players converted their kicks. Kelleher scored his kick before his counterpart Kepa, who ironically had been brought off the bench specifically for the shoot out, blasted over the bar.

In a third round Carabao Cup against Derby County on 9 November 2022 Kelleher saved three penalties in a shoot out victory. This made him Liverpool's most successful keeper in penalty shoot outs with six saves. He was limited to four appearances in all competitions however and at the end of the season his national team manager Stephen Kenny predicted he would be moving on from Liverpool to ensure more game time. That didn't happen and he remained with the Reds for 2023/24, making 26 appearances including ten in the Premier League due to injuries to Allison Becker.

Kelleher’s greatest asset that is so evident is his composure. Rarely flustered, even in the most intense of situations, he has built a reputation for staying calm under pressure – most memorably in penalty shoot-outs, where his timing and confidence have often proved decisive. Technically, he is an agile shot-stopper with quick reflexes, capable of spreading himself to make important saves at close range. His background as an outfield player is evident in his distribution: he is comfortable with the ball at his feet, able to play accurate passes into midfield or switch play to the flanks, and fitted neatly into Liverpool’s style of building from the back. Another key strength is his mentality. Despite long spells on the bench behind Alisson, Kelleher maintained sharpness and professionalism, stepping in seamlessly whenever called upon. His reliability in high-stakes cup matches made him a trusted figure in the squad. He also commanded respect for his bravery, willing to come off his line to claim crosses or close down attackers, and for his steady temperament that spreads confidence to the defence in front of him. 

Liverpool purchased Giorgi Mamardashvili from Valencia on 27 August 2024, but would remain on loan at Valencia through the 2024/25 season. Kelleher's entry into the first team seemed further complicated as he noted in the summer of 2024: "I've made it clear in the last few years that I want to go and be a No. 1 and play week in, week out," Kelleher said while on international duty with Ireland. The club's made that decision to get another goalkeeper, and from the outside looking in, it looks like they've made a decision to go in a different direction. “I want to go somewhere and be a number one. Liverpool rejected a few bids as well. It’s not always in my hands to make the decision... My ambition is clear. I’m good enough and I want to go and play week in, week out.” Despite his frustrations, Kelleher had to bide his time once more, staying on for another campaign that always looked destined to be his last at Anfield.

During 2024/25 another injury to Allison meant Kelleher enjoyed a run of ten straight Premier League and Champions League games in the autumn. He didn't let anyone down as always and even saved a penalty from Kylian Mbappé in a 2-0 victory over Real Madrid at Anfield. However, he was only keeping the gloves warm for Alisson's return and with the impending arrival of Marmardashvili, it became clear Kelleher needed to move elsewhere if he was to become a first choice keeper. Kelleher joined Brentford in a transfer deal worth up to £18 million on 3 June 2025. The initial fee is £12.5 million, with performance-related add-ons potentially increasing the total amount. Ringmahon Rangers, who sold Kelleher to Liverpool in 2015 for £30,000, will receive £2.5 million from the transfer fee thanks to a sell-on clause included in the original deal.