Born in Hartlepool, County Durham, he studied at Wrexham School of Art, North East Wales, then Denbigh Technical College and later the University of Reading from 1960–64. In Liverpool, where he lived for nearly twenty years from 1964, he taught at Liverpool College of Art and Liverpool Polytechnic and Nottingham University. He was a central figure in Liverpool's artistic life, regularly exhibiting at the Walker Art Gallery, before his departure for London in 1982. Cockrill's Liverpool work was in line with that of John Baum, Sam Walsh and Adrian Henri, employing Pop and Photo-Realist styles, but later he moved towards Romantic Expressionism, as it was shown in his retrospective at the Walker Art Gallery, Liverpool in 1995. His poetry was published in magazines such as "Ambit" and "Poetry Review".
He was formerly the Keeper of the Royal Academy, and as such managed the RA Schools of the Establishment as well as being a member of the Board and Executive Committee.
Cockrill received the John Moores Liverpool Exhibition, in addition to awards from the Arts Council of Great Britain, Flags and other Projects, Royal Festival Hall, the Arts Council of Great Britain (Major Award), and the Arts Council Works of Art in Public Spaces. He was also given a British Council Award in 1985.