He briefly attended the Regent Street Polytechnic before entering a variety of jobs, including troubleshooter at a TV and radiogram manufacturer and as a negotiator in a boutique West End estate agency, before applying for drama school.

He acted alongside Elaine Stritch in The Gingerbread Lady, Denholm Elliott in The Paranormalist, Frank Langella in Abracadaver, Albert Finney in Ronald Harwood's Reflected Glory, Felicity Kendal and Frances de la Tour in Fallen Angels, Joseph Fiennes in George Dillon and Lesley Manville in Six Degrees of Separation.

From 2016 to 2018, together with Fenella Fielding, Greif appeared at Crazy Coqs at Zedel's Brasserie on multiple occasions, and various other venues in David Stuttard's adaptation of drama and comedy from famous ancient Greek literature called Tears, Treachery..and Just a little Murder produced by Simon McKay.

Greif appears in many films including Gerry, Risen, Woman in Gold, Bill, D is for Detroit, Lasse Hallström's Casanova, Shoot on Sight, Eichmann, Spartan, The Upside of Anger, Boogie Woogie and Fakers.

His many television appearances include Only When I Laugh, Thriller (1 episode, 1976), Howard's Way, Spooks, Mistresses, He Kills Coppers, Silent Witness, the last story of Waking the Dead, The Crown, The Alienist: Angel of Darkness, Doctors and New Tricks.