Chevron grew up in Santry, a suburb of Dublin. Beginning in the late 1970s, he was lead singer and co-founder of the punk rock group The Radiators from Space, receiving some critical acclaim but little widespread popularity or financial success. Following a temporary breakup of the band in 1981, he lived in London for a while, meeting and befriending Shane MacGowan through time spent working together at a record shop. Following the release of the Pogues' 1984 debut album "Red Roses For Me", he was invited to join the band on a short-term basis as cover for banjo player Jem Finer's paternity leave. He took over as guitarist following MacGowan's decision to concentrate on singing—thereby becoming a full-time member of the band in time for the recording of its second album, "Rum, Sodomy and the Lash".