Claude Vivier was born to unknown parents in Montreal and adopted at the age of three by a poor French-Canadian family. From the age of thirteen, he attended boarding schools run by the Marist Brothers, a religious order that prepared young boys for a vocation in the priesthood. The young Vivier's religious inclinations were supplanted by a love of modern poetry and music. At the age of eighteen, Vivier was asked to leave the novitiate and the following year enrolled in the Conservatoire de musique du Québec à Montréal, where his main teacher was the composer Gilles Tremblay. His earliest works date from this period.
In 1971, he began a period of three years' study in Europe, first at the Institute for Sonology in Utrecht, and then in Cologne with Karlheinz Stockhausen. Vivier learned much from Stockhausen, and his early works have aspects that are clearly, and sometimes audibly, derivative of his teacher, even though his later works bear little audible resemblance. In 1974, he returned to Montreal and began to establish his reputation. Early works like Lettura di Dante were performed with some success at the concerts of the SMCQ. In autumn 1976, he undertook a long trip to the East, notably to Japan and Bali. The music he encountered there had a profound effect on him. Subsequent compositions like Pulau Dewata show the impact quite audibly, but in later works the influence has been digested and goes much deeper.
Vivier's opera Kopernikus, to his own libretto, was premièred on 8 May 1980, at the Monument-National in Montreal. By that time he had begun to compose in a somewhat different manner, influenced by the techniques of French spectral music, and was notably influenced by Gérard Grisey and Tristan Murail. One of the first of the works in this new manner, Lonely Child (1980) for soprano and orchestra, has become his best-known work. This and other late scores of Vivier, including Prologue pour un Marco Polo and Wo bist du Licht! were intended for inclusion in an unfinished "opéra fleuve" entitled Rêves d'un Marco Polo.
In June 1982, with the help of a Canada Council grant, Vivier left Montreal for Paris, where he began work on an opera based on the death of Tchaikovsky. In March the following year, he was stabbed to death by a young Parisian man who may have been a prospective lover and who was later caught and sentenced. His last work was the unfinished Glaubst du an die Unsterblichkeit der Seele, which contains a disturbing premonition of his death.