Copley was born in Denby Dale, West Yorkshire, and grew up beside a dairy farm there. His father, Harold, was involved with local amateur dramatic productions, as were the rest of his family. He went to Penistone Grammar School, then the Northern Counties College of Education (now Northumbria University) in Newcastle upon Tyne, where he received an Associate of the Drama Board (ADB) in Drama. He taught English and Drama in Walthamstow, before he joined the Leeds Playhouse Theatre-in-education Company in 1971.
Copley played a small role as Private Wicks in the 1977 film 'A Bridge too far. In 1979 he played another small but noticeable role in Zulu Dawn as Cpl Storey in the British Army. He has played Matthews in Hornblower, Ian in Roughnecks and Jerry in This Life and Peter Quinlan in The Lakes. He has appeared in Queer as Folk as Nathan Maloney's father, Big Finish Productions's July 2002 Doctor Who story Spare Parts and in Shameless as a water sports enthusiast. In 1980 he appeared in critically acclaimed drama Minder playing George Palmer in episode The Old School Tie. He narrates the Channel 4 programme, How Clean is Your House?. He featured in the ITV children's hit show "Best Friends" 5 episodes in 2005-2006, playing the grandfather.
He is a noted perennial BBC Radio 4 radio play actor, usually in gritty or romantic plays about hard-working folk set in the north of England. They are often repeated on BBC7. Whenever a genial Yorkshire accent has been cast in the BBC radio drama] department, he has often been summoned. He narrated the 1985 Yorkshire Television 9 part serial adaptation of The Pilgrim's Progress entitled Dangerous Journey.
On 13 February 2006, Copley appeared as an angry hostage-taker in an episode of the crime drama Life on Mars. Copley appeared in the TV Soap Coronation Street on 8 August 2007, portraying a character called Ivor Priestley, and in the TV adaptation of The Worst Witch by Jill Murphy, as wizard and former-frog, Algernon Rowan-Webb.
In 1976, he won the Laurence Olivier Award for Best Actor in a New Play for his role in John Wilson's For King and Country.
In 2009 he appeared in the third series of BBC One show Torchwood.
In 2010 he appeared in an episode of BBC One show Casualty and Survivors.
In 2011 he appeared as William Mason's father in several episodes of Downton Abbey, and in 2012played Alan in White Heat.
He married the actress Natasha Pyne in 1972, after performing with her in a Leeds Playhouse production of Frank Wedekind's Pandora's Box, adapted by Peter Barnes, directed by Bill Hays in 1971.