As you walk into the theatre you are greeted by the sight of hi-vis jacketed workman, digging beneath the stage. The scene is a car park in Leicester, and before the action of Shakespeare's sprawling history opens, we see archaeologists uncover Richard III's grave – holding the bones of a curving spine to the light.

Then, out of the darkness at the back of the stage, a halting crook-backed figure emerges. Past and present come together, fiction and history, and Ralph Fiennes' Richard stands before us, leaning on a sword-stick, dressed in black polo neck and neat black blazer, a timeless embodiment of the evil that men do.

As he begins that famous first speech "Now is the winter of our discontent" I felt a thrill. Each line is new-minted, each thought seems to spring from the moment not the page. In civilised, gentle tones Fiennes finds new sense in the old words. This is a mesmeric, revelatory performance, making Richard a man who masks his intent under the demeanour of a civil servant – but whose eyes flash daggers.

He can be funny but he is also utterly terrifying. When he suddenly turns on Hastings, thrusting his withered arm in front of the frightened courtier, or when he can barely stop himself from running a sword through one of the annoying young princes, or – most shockingly – when he rapes Queen Elizabeth as a prelude to wooing her daughter, we see this Richard in all his foul fury. But it is a performance of great subtlety too. Richard's hatred is perhaps driven by a loathing for women; Fiennes gives a visible start when Finbar Lynch's Buckingham describes him as "effeminate."

This towering creation sets the tone of Rupert Goold's mainly modern dress production, more sombre, more serious than sometimes, but more pertinent too. Watching it on the day of MP Jo Cox's shocking death, it was impossible not to feel the links between past and present; the way moral ugliness can sometimes prevail. Hildegard Bechtler's set crowns the action with a great circle of light – perhaps the swirl of the car park – that highlights the actors' faces, and sets their deeds into shadow. Skulls on the back wall gleam to record Richard's many victims.

Every scene and each character attains a timeless force. James Garnon's self-satisfied Hastings blunders around refusing to recognise the signs of danger; Lynch's Buckingham gleams with steely ambition; Mark Hadfield is convincing both as a local mayor out of his depth among the big boys, and a servant trying to work out the will of his master.

Then there is Vanessa Redgrave, making her Almeida debut at the age of 79, and turning the curses of Queen Margaret into the meanderings of a woman driven mad by grief. She enters slowly, clutching a doll, symbol of all the children who have been killed in battle. When, near the close, she sits on the ground to mourn with Susan Engel's Duchess of York and Aislín McGuckin's grieving Elizabeth, you see women who have fought, lost and realised the cost. It is deeply affecting.

This has been a year of Richard III - Goold's is the third production I've seen and my favourite. Its dark anger is compelling, making the voices of the past speak loudly from the grave. It's on in cinemas on July 21. See it if you can.

Richard III runs at the Almeida Theatre until 6 August 2016.