In 1960 Junkin joined Joan Littlewood's Stratford East Theatre Workshop and played the lead in the original production of Sparrers Can't Sing. A few years later he joined the Royal Court Theatre company and was the foil to Tony Hancock in some of Hancock's last work for British television. He played a diverse range of roles on the small screen; however, he is best remembered for his comedy roles and his appearances as a television quiz master. Worldwide filmgoers will remember him best for playing "Shake", the assistant to Norman Rossington, in the Beatles film A Hard Day's Night. In comedy roles, Junkin was rarely short of work, on account of his outstanding ability to play the stony-faced symbol of low level, petty-minded and unquestioning authority, whether the army sergeant, police constable or site foreman.

One of his rare leading roles was in the BBC series The Rough with the Smooth, in which he and Tim Brooke-Taylor played comedy Writers (with both actors contributing scripts to the series as well). He also hosted his own afternoon television series in the mid-1970s. Entitled simply Junkin, it was produced by Southern Television for the ITV network.

Junkin lived in Wendover, Buckinghamshire. He married Jenny Claybourn in 1977 and had one daughter. He died on 7 March 2006 in the Florence Nightingale House, Aylesbury, several miles from his home. He had been suffering from lung cancer, emphysema and asthma. His life and work were honoured at the British Academy Television Awards in 2006.