Damonte spent most of his youth in Montevideo. His maternal grandfather was the journalist Natalio Félix Botana and his father was the journalist Raúl Damonte Taborda, an antiperonist Radical politician and director of the journal Tribuna Popular. Raúl showed an early talent for drawing and, from his adolescence, contributed caricatures to his father's publication and to the satirical magazine Tía Vicenta.

His father's political activities forced the family into exile in Uruguay, Haiti, and later New York. He finally settled in Paris, where he embarked on a career as a cartoonist for such newspapers as Le Nouvel Observateur. His most notable character during this period was La Femme assise, The Sitting Woman.

He was a member of Tse, an association of Franco-Argentine artists with whom in 1969 he staged a biographical play about Eva Perón. His theatrical works, influenced by Samuel Beckett, are characterized by the failure of characters to communicate.

Copi also collaborated with the avant-garde group Pánico, which included Fernando Arrabal, Roland Topor, and Alejandro Jodorowsky.

Copi contributed cartoons to the magazine Gai Pied.

He died of an AIDS-related illness in 1987, at the age of 48.

The Argentine writer César Aira, who wrote an essay on Copi, has given lectures on his works in the Ricardo Rojas Cultural Center and is credited with popularizing Copi in Argentina. His posthumous reputation is based largely on his plays, stories, and novels.