Hopkins was born in Margam, Port Talbot, Wales, on 31 December 1937, the son of Muriel Anne (née Yeats) and Richard Arthur Hopkins, a baker. His schooldays were unproductive; he found that he would rather immerse himself in art, such as painting and drawing, or playing the piano, than attend to his studies. In 1949, to instill discipline, his parents insisted he attend Jones' West Monmouth Boys' School in Pontypool, Wales. He remained there for five terms and was then educated at Cowbridge Grammar School in the Vale of Glamorgan, Wales.
Hopkins was influenced and encouraged to become an actor by Welsh compatriot Richard Burton (who was also born in Port Talbot), whom he met briefly at the age of 15. To that end, he enrolled at the Royal Welsh College of Music & Drama in Cardiff, Wales, from which he graduated in 1957. After two years in the British Army doing his national service, he moved to London, where he trained at the Royal Academy of Dramatic Art.