Born in Leytonstone, London, Robinson attended Wanstead High School in what is now the London Borough of Redbridge. His first professional acting role came at the age of 12, as a member of Fagin's gang in the original production of the musical Oliver!, including a stint as the Artful Dodger when the boy playing the role didn't turn up. He then spent five years during which he appeared in a number of West End shows, in film, and on television.

At school, Robinson passed four O Levels (English Language, English Literature, History and Geography) and went on to study for A Levels, but decided to study at a drama school instead. Too young to attend the Royal Academy of Dramatic Art, Robinson studied at the Central School of Speech and Drama. After leaving, he spent four years in repertory theatre.

He won an Arts Council bursary to work as a director at the Midland Arts Theatre, Birmingham and founded the Avon Touring Company, a Bristol-based community theatre company, with writer David Illingworth.

He appeared in the 1974/75 season at Chichester Festival Theatre, as 'Angel Chicago' in the nativity musical Follow The Star. In the 1975 season, he appeared as Hovstad in Ibsen's Enemy of the People. In 1976, he appeared as Feste in Twelfth Night, and as Majorin in Monsieur Perrichon's Travels.

In the early 1970s he starred in the children's educational programme Sam on Boff's Island, which was about a man who falls asleep while eating his breakfast cereal and ends up transported to an island which looks just like his cereal. He also appeared in the award-winning Horizon documentary Joey, and in the title role in the BBC production of The Miracle of Brother Humphrey. He also appeared in the film Branningan alongside John Wayne.

He was also one of the Who Dares Wins team in the ground-breaking Channel 4 comedy/satirical show during the early/mid-1980s.

Robinson came to prominence in 1983 for his role in the British historical sitcom Blackadder, as Edmund Blackadder's dogsbody Baldrick. In the first series, broadcast as The Black Adder, he was quite astute, whilst his master was an idiot.

Later series (Blackadder II, Blackadder the Third, Blackadder Goes Forth) moved the duo through history and switched the relationship: the Edmund of Blackadder II was an Elizabethan schemer, whereas Baldrick had devolved into an intellectually challenged buffoon whose catch phrase was "I have a cunning plan" (which he never did).

His shortness (5' 4" (1.63 m)) was especially apparent next to his tall co-stars (Rowan Atkinson 5' 11", Hugh Laurie 6' 2.5", Stephen Fry 6' 5"), so Baldrick was frequently the butt of jokes.

In addition to his acting on Blackadder, he also wrote and narrated several Jackanory-style children's programmes, encouraged by Richard Curtis. These programmes would see Robinson bounding about the place acting out all the parts.

Programmes in this style included Tales From Fat Tulip's Garden (continued in Fat Tulip Too), Odysseus: The Greatest Hero of Them All (a retelling of the Iliad and the Odyssey), and Blood and Honey (tales from the Old Testament, filmed on location).

In the late 1980s he created the children's comedy TV series Maid Marian and her Merry Men, a loose retelling of the legend of Robin Hood in which he appeared as the Sheriff of Nottingham.

Four series were broadcast on BBC1 during 1989-94. With its modern spin on ancient times, the hugely popular series was considered by many to be a junior equivalent of Blackadder.