Crozier was born in Glasgow to Irish parents and educated at the Glasgow School of Art between 1949 and 1953. On graduating he spent time in Paris and Dublin before settling in London, where he gained a reputation as the 1950's equivalent of a Young British Artist through the early success and notoriety of his exhibitions of assemblages and paintings at the ICA, Drian and the Arthur Tooth galleries, with whom he had a long association.
Profoundly affected by post-war existential philosophy, Crozier allied himself and his work consciously with contemporary European art throughout the 1950's and 1960's, rather than with the New York abstractionists, who were more fashionable in the UK at the time. He was also part of the artistic and literary world of 1950's Soho, a close associate of 'the Roberts', Colquhoun and MacBryde, John Minton and William Scott, and part of the expatriate middle-European and Irish intellectual circles in London of the time. Crozier spent 1963 in southern Spain with the Irish poet Anthony Cronin; this proved pivotal to Crozier's development as an artist. On his return to the UK, he began a series of skeletal paintings which anticipated the 'New Expressionist' German painters of the 1980's, and which were influenced by Crozier's visit in 1969 to Auschwitz and Belsen.
Based in London throughout the 1960's and 1970's, Crozier exhibited his works in London, Glasgow, Dublin and all over Europe. As many artists of the 1960's did, Crozier combined painting with teaching, first at Bath Academy of Art, (with Howard Hodgkin, Gillian Ayres and Terry Frost), then at the Central School of Art (with William Turnbull and Cecil Collins), at the Studio School in New York and finally at Winchester School of Art where he led a strong centre for painting based on the European tradition. From the 1980's, Crozier's painting blossomed with a new freedom and confidence, the result of his giving up teaching and the stimulus provided by his studios in West Cork in Ireland, and in Hampshire in England. His abstract landscapes and still life painting used sumptuous colour to convey an emotional intensity. To the end of his life, he was endlessly concerned with the challenge of creating a new language in figurative painting.