Rodway was born in Dublin to English parents, Frank and Lillian (Moyles) Rodway. He studied at Trinity College, Dublin, where he was elected a scholar in classics in 1948. He worked as an accountant, Teacher and university lecturer before acting.

He made his stage debut in May 1953 at the Cork Opera House. There, he portrayed General Mannion in The Seventh Step. He made his first appearance in London in 1959, as The Messenger in Cock-A-Doodle Dandy. In 1962, he portrayed the young James Joyce in Stephen D, based on Joyce's writings. Rodway joined the Royal Shakespeare Company in 1966. His favourite theatrical parts included Bassov in Summerfolk, and the title roles in Butley and Richard III.

Although he was primarily a stage actor, he also performed in radio, television and film productions. With his expressive voice (described by Jack Adrian as "rich and dark and thumpingly Celtic" ), he made many radio broadcasts for the BBC. Major television roles included Cummings in Reilly, Ace of Spies, and Charles Brett in The Bretts. He also appeared in series such as Miss Marple, Rumpole of the Bailey and Inspector Morse. He acted with Orson Welles in Chimes at Midnight (1965), I'll Never Forget What's'isname (1967) and Patrick McGoohan in an episode of Danger Man; "The Man Who Wouldn't Talk". He often acted as the villain, including Adolf Hitler in The Empty Mirror (1999). He played the role of Apemantus in both television and audiobook productions of Shakespeare's Timon of Athens.

He was married four times. His first wife was the Actress Pauline Delaney, and his second the casting Director, Mary Selway. He was stepfather to Tara FitzGerald by his third marriage to Sarah Callaby (née Fitzgerald); they had a daughter, Bianca. He was married to Jane Rodway from 1991 to his death. He died in London after a series of strokes.