Born in 1980 or 1981, Clayton became involved in music as a chorister at Worcester Cathedral. His voice changed late: he has said that at 15, he could still sing the high C in Gregorio Allegri's Miserere. Clayton earned a choral scholarship to St. John's College, Cambridge, also studying social anthropology while there and starring in the lead role in a Cambridge University Opera Society production of Britten's Peter Grimes. He went on to the Royal Academy of Music, earning the school's first Sir Elton John Scholarship and winning the Queen's Commendation for Excellence. Clayton earned widespread attention as a BBC New Generation Artist from 2007 to 2009 and expanded his skills with a fellowship from the Borletti-Buitoni Trust beginning in 2008. That year, he made his debut in the title role of Britten's Albert Herring at the Glyndebourne Festival. His song recital debut at Wigmore Hall in London, featuring songs by Schumann, Wolf, and Tippett, received wide attention. Since then, Clayton has appeared with such companies as the Komische Oper in Berlin, the Royal Opera Covent Garden in London, and the Bayerische Staatsoper in Munich. He has a vigorous schedule of recitals with a variety of accompanists, including Graham Johnson, Julius Drake, and Malcolm Martineau. Clayton's concert appearances include those with the BBC Symphony Orchestra, the Gürzenich Orchestra of Cologne, and Les Violons du Roy. When the coronavirus epidemic broke out in early 2020, he was slated to appear in a new Covent Garden production of Janáček's Jenufa.
In addition to opera recordings, Clayton has appeared on various other albums beginning with a solo part in the 11-year-old Mozart's Die Schuldigkeit des ersten Gebots, K. 35, on a Signum Classics recording in 2013. He recorded the album Where'er You Walk: Arias for Handel's Favourite Tenor for Signum Classics in 2016. He has also recorded for Chandos and Hyperion. Clayton was heard with Mary Bevan and the Carducci String Quartet on the album Ian Venables: Love Lives Beyond the Tomb - Songs and Song Cycles.