Clore received his undergraduate degree with first class honours in biochemistry from University College London in 1976 and medical degree from UCL Medical School  in 1979. After completing house physician and house surgeon appointments at University College Hospital and St Charles Hospital (part of the St. Marys Hospital group), respectively, he was a member of the scientific staff of the Medical Research Council National Institute for Medical Research from 1980 to 1984. He received his PhD from the National Institute for Medical Research in Physical Biochemistry in 1982. He was awarded a joint Lister Institute Research Fellowship from the Lister Institute of Preventive Medicine which he held from 1982 to 1984 at the Medical Research Council. In 1984 he joined the Max Planck Institute for Biochemistry in Martinsried, Germany, where he headed the Biological NMR department from 1984 to 1988.
In 1988, Clore was recruited to the National Institutes of Health (NIH) Laboratory of Chemical Physics (National Institute of Diabetes and Digestive and Kidney Diseases) located in Bethesda, Maryland, U.S., where he interacted closely in the late 1980s and early 1990s with NIH colleagues Ad Bax, Angela Gronenborn and Dennis Torchia on the development of multidimensional heteronuclear NMR spectroscopy and a structural biology effort aimed at proteins involved in the pathogenesis of HIV/AIDS.  He has remained at the NIH ever since and is currently a NIH Distinguished Investigator and Chief of the Section on Molecular and Structural Biophysics at the NIH. He is an elected Member of the United States National Academy of Sciences, a Fellow of the Royal Society, a Fellow of the American Academy of Arts and Sciences, and a Foreign Member of the Academia Europaea (Biochemistry and Molecular Biology Section). Clore's citation upon election to the Royal Society reads: