Armstrong made his debut in the 1971 film Get Carter, and has appeared in several films, although in this medium he has usually played supporting or minor roles. He has appeared in Sleepy Hollow, The Mummy Returns and Van Helsing. He appeared as Torquil, the bandit leader who agrees to help Colwyn defeat the evil Slayers, in the 1983 movie Krull.

On television, Armstrong has played the character of Brian Lane in the highly popular BBC drama New Tricks where he was reunited with James Bolam, with whom he appeared in an episode of Whatever Happened to the Likely Lads? called "Conduct Unbecoming" and the feature film The Likely Lads, in which he played Terry's milkman. Armstrong also starred in the 1981, Yorkshire Television drama, Get Lost!, which inspired the later better known drama, The Beiderbecke Affair, in which James Bolam played Armstrong's counterpart.

He also played Austin Donohue in the 1996 BBC drama series Our Friends in the North, a character based on the politician T. Dan Smith, Inspector Bucket in the 2005 BBC adaptation of Bleak House and Mr Evans in Carrie's War. In 2001, Armstrong played George Mole in the television series Adrian Mole: the Cappuccino Years In 2003, Armstrong starred alongside his son, Joe Armstrong, in the ITV1 drama Between the Sheets.

In 2009, he appeared as Jeremiah Flintwinch in Little Dorrit. Armstrong also played the character of John Southouse in the BBC period legal drama Garrow's Law in the same year.

Armstrong has worked regularly in British stage and television productions, such as Granada Television's The Stars Look Down (1975).

Armstrong spent nine years with the Royal Shakespeare Company at Stratford upon Avon and London, including the roles of Mr Squeers in Nicholas Nickleby, Petruchio in The Taming of The Shrew, Leontes in The Winter's Tale, Barabas in The Jew of Malta and Monsieur Thénardier in the original cast of Les Misérables. In 2006, he returned to the London stage to star in Trevor Nunn's new production of The Royal Hunt of the Sun at the National Theatre.

Later in 2006 he starred as Uncle Garrow in the film Eragon. He has been nominated for the Laurence Olivier Award six times and won in 1994, for Best Actor in a Musical for his performance of Sweeney Todd in the musical of the same name.

Armstrong is well known for his performance as Thénardier in the original London production of Les Misérables in October 1985 with Susan Jane Tanner alongside him as Mme. Thénardier, for which he was also nominated for a Laurence Oliver Award for Best Actor in a Musical. He reprised this role in Les Misérables - The Dream Cast in Concert, at the Royal Albert Hall in October 1995.

On 15 July 2009, Armstrong was awarded an honorary Doctorate by the University of East Anglia in recognition of his work, and was the very next day awarded another honorary Doctorate from the University of Sunderland in recognition of outstanding contributions to television and film.

He is the father of actor Joe Armstrong, who found fame in the current BBC version of Robin Hood.