Acosta was born to Jose Acosta and Alexandrina Perez de Acosta on July 29, 1920 in the disputed American territory of Chamizal outside of El Paso, Texas. His father, a carpenter, moved the family to Los Angeles, where Acosta was raised and graduated from Lincoln High School. Acosta studied drama at Los Angeles City College and UCLA and he appeared at the Pasadena Playhouse. At the age of 19, he received a scholarship to the Palacio de Bellas Artes in Mexico City where he studied for three years. In 1943, during World War II, Acosta enlisted in the United States Navy where he worked in Naval Intelligence.