Murphy was born in Ventnor, Isle of Wight. He was called up to do his national service at RAF Northwood, where he met future The Good Life actor Richard Briers. On leaving the RAF the two aspiring actors both performed in productions by the Dramatic Society at the Borough Polytechnic Institute, now London South Bank University. Murphy was a member of Joan Littlewood's Theatre Workshop (alongside Yootha Joyce), and a jobbing actor in the 1960s and early 1970s, combining his theatre work with appearances in television shows such as The Avengers, Z-Cars, Callan (Reeves in 'None of Your Business', 1972) and Dixon of Dock Green, before being cast in the role which would make him a household name.
In Man About The House, Murphy played the lazy and scheming George Roper, the landlord of a house converted into flats who had a young man and two women renting the flat above. His wife, Mildred, was played by Yootha Joyce. Her domineering, social-climbing characteristic was a sharp contrast to George's desire for an easy and quiet life and the pairing was an instant hit. A contributing factor to their immediate chemistry was the fact they had been friends for many years. Aside from their Theatre Workshop years, Murphy also featured in Sparrows Can't Sing (1963) with Joyce and other Theatre Workshop colleagues. When Man About The House ended in 1976, a spin-off was created for Murphy and Joyce, entitled George and Mildred. This ran for five series until 1979. Murphy reprised his role in feature films of both sitcoms. Joyce died in 1980.
Murphy has continued to appear regularly on television, most notably as "Alvin Smedley" in Last of the Summer Wine from 2003 to 2010, and also with memorable roles in On Your Way, Riley (1985) as Arthur Lucan, Lame Ducks (as a private investigator); The Bill(as a drunken tramp dressed as an elf at Christmas); comedy series' Pond Life, an animation series, as Len Pond, the father of protagonist Dolly Pond (Sarah Ann Kennedy), One Foot in the Grave starring Richard Wilson (playing a character called Mr Foskett);,Brookside; and as a shopkeeper called Stan in Wizadora. In 2010 he appeared in an episode of Hustle called "The Thieving Mistake". He has more recently made appearances in comedy shows The Catherine Tate Show, This is Jinsy and Benidorm.