Born in Stockton-on-Tees and raised in Thornaby, Griffiths was the son of deaf parents and he learned sign language before he could speak. He experienced a somewhat troubled childhood, which included frequent attempts to run away from home, and he dropped out of school at age 15. Hired as a porter, he was encouraged to return to school by his boss and a drama class at Stockton & Billingham College literally changed his life.

Shortly after leaving the college, the portly actor earned a spot in the repertory company of BBC Radio. Too young to be a character player and too hefty to be a young leading man, Griffiths then found himself working in small theaters around Britain, sometimes acting, sometimes stage managing. Finally settling in Manchester, he began to get solid parts on stage, including in an early work of a then unknown Alan Ayckbourn. Griffiths also made his first forays on the small screen, appearing in bit roles in Granada Television productions. By chance, he was working in front of the cameras when Trevor Nunn, the artistic director of the Royal Shakespeare Theatre Company, happened to be in the studios and saw him on a monitor. Nunn encouraged the actor to head to London for a spot with the Royal Shakespeare Company. Griffiths spent several seasons with the RSC at first playing small comic roles in the classics, such as Peter in "Romeo and Juliet" and six lines in "A Comedy of Errors". Proving the old adage that there are no small parts, Griffiths proved a success with audiences despite his limited stage time and was graduated to more important roles like Bottom in a 1977 staging of "A Midsummer Night's Dream". In 1979, he earned several accolades for his turn as the Hollywood-bound George in the Kaufman and Hart play "Once in a Lifetime". He went on to be perfectly cast as Falstaff in "The Merry Wives of Windsor" (filmed for British TV) and appeared in several other productions as well.

Having made his feature debut in "It Shouldn't Happen to a Vet/All Things Bright and Beautiful" (1975), Griffiths landed the key role of the British husband of an American woman (Elaine Stritch) in "Nobody's Perfect" (1980, 1982), a sort of Anglicized version of the popular American sitcom "Maude". The one-time porter portrayed one in the Oscar-winning "Chariots of Fire" (1981) and appeared as a reporter in the next year's Best Picture "Gandhi". He excelled as a mild-mannered computer genius who discovers a financial conspiracy in the thriller "Bird of Prey" (1982) and reprised the part in a 1984 sequel "Bird of Prey 2". In between, Griffiths had a chance to make an impression with audiences as a cynical lawyer in the feature "Gorky Park" (1983) and as the accountant who inexplicably falls in love with a pig in "A Private Function" (1985). But it was his turn as gay Uncle Monty with eyes for Paul McGann's Marwood (the "I" of the title) in "Withnail and I" that really established him as a character player in films, although he retreated to British sitcoms ("Ffizz", ITV 1987, 1989; "A Kind of Living", ITV 1988-90; "Pie in the Sky", BBC 1994-97). The actor did make occasional appearances in TV-movies that aired in the USA like "Casanova" (ABC, 1987) and features like "Guarding Tess" (1994), in which he was dryly amusing as a butler. In 2000, Griffiths delivered a finely calibrated turn as the piggish thug overseeing the kitchen in the BBC miniseries "Gormenghast" and appeared on the London stage the following year in the long-running comedy "Art" and the revival of "Luther" starring Rufus Sewell. After earning his stripes as Uncle Vernon Dursley in the first Harry Potter film, the actor reprised the role in the sequel "Harry Potter and the Chamber of Secrets" (2002).