Born in Derby, Derbyshire, England, Dexter left school at the age of fourteen to serve in the British Army during the Second World War.
Dexter continued with Hamlet (with music by Conrad Susa, 1969), Equus (one of his triumphs, 1973), Trevor Griffiths's The Party (Lord Olivier's final stage appearance, 1973), Phaedra Britannica (with his friend, Diana Rigg, 1975), The Merchant (aka, Shylock, 1977), As You Like It (with music by Harrison Birtwistle, 1979), Life of Galileo (with Sir Michael Gambon, 1980), The Glass Menagerie (with Jessica Tandy, 1983) and Julius Caesar (1988).
Dexter made his operatic debut at the Royal Opera, Covent Garden, in 1966, with Benvenuto Cellini, with Nicolai Gedda, and, in 1983, he staged a double-bill of Le rossignol (with Natalia Makarova) and L'enfant et les sortilèges for the company.
At the Hamburg State Opera, he staged Les vêpres siciliennes (1969), From the House of the Dead (1972), Billy Budd (with Richard Stilwell, 1972), Boris Godunov (in the Shostakovich edition, 1972) and Un ballo in maschera (with Luciano Pavarotti and Sherrill Milnes, 1973).
At the Metropolitan Opera (where he was Director of Production from 1974 to 1981, then Production Advisor from 1981 to 1984), Dexter mounted new productions of Les vêpres siciliennes (with Montserrat Caballé, 1974), Aïda (with Leontyne Price, 1976), Le prophète (with Marilyn Horne, 1977), Dialogues des Carmélites (1977), Lulu (1977), Rigoletto (with Cornell MacNeil and Plácido Domingo, 1977), Billy Budd (with Sir Peter Pears, 1978), The Bartered Bride (with Teresa Stratas and Jon Vickers, 1978), Don Pasquale (with Beverly Sills, 1978), Don Carlos (with Renata Scotto, 1979), Die Entführung aus dem Serail (with Edda Moser, later Zdzisława Donat, 1979), Aufstieg und Fall der Stadt Mahagonny (1979) and the triple-bills (both designed by David Hockney) of "Parade" (Parade/Les mamelles de Tirésias/L'enfant et les sortilèges, 1981) and "Stravinsky" (Le sacre du printemps/Le rossignol/Œdipus rex, 1981).