Cranham trained at the National Youth Theatre of Great Britain and RADA. He starred in the title role in the popular 1980s comedy drama Shine on Harvey Moon. He also appeared in Layer Cake, Gangster No. 1, Rome, Oliver! and many other films. Cranham was cast as the deranged Dr. Philip Channard and his Cenobitic alter-ego in the Horror film Hellbound: Hellraiser II. Among many stage credits are West End productions of Entertaining Mr Sloane, Loot, An Inspector Calls (both transferring to Broadway), The Ruffian on the Stair, The Birthday Party and Gaslight (at the Old Vic). For his role as Inspector Goole in An Inspector Calls, he was nominated for a Laurence Olivier Award.

For BBC Radio 4's Afternoon Play, Cranham played DS Max Matthews in the three-play series The Interrogation by Roy Williams and starred as Thomas Gradgrind in BBC Radio's 2007 adaptation of Dickens' Hard Times.

In 2016, Cranham won the Olivier Award for Best Actor in a Play for his role as Andre in Florian Zeller's The Father. The play originated at the Theatre Royal Bath's Ustinov Studio in the Autumn of 2014, before touring the country and transferring to the West End in the Summer of 2015, and returning to the Duke of York's Theatre in Spring 2016. The play received an unprecedented five star review from every leading national press publication, and Cranham's performance was described as "the performance of his life" (WhatsonStage.com).