In the 1930's, Hector's parents, Martin and Carmen Elizondo migrated to New York, like so many other Puerto Ricans at the time, with the hope of finding a better way of life. They had two children, Hector being the first. At a young age, Elizondo demonstrated that he was talented in both sports and music. When he was 10 years old, he sung for the "Frank Murray Boy's Choir" Upon graduating from Jr. High School in 1951, he enrolled in the High School of the Performing Arts. Elizondo also attended a regular public high school where he excelled in basketball and baseball. His baseball skills were good enough for him to be scouted by the New York Giants and the Pittsburgh Pirates.
In 1954, Elizondo enrolled in New York City College, with the intention of becoming a history teacher. During his freshman year he became a father. He dropped out of College and went to work full-time in order to support his family. Later, he was divorced and gained full custody of his son, Rodd.
From 1962-63, Elizondo studied dance at the Ballet Arts Company at Carnegie Hall. In 1963, Elizondo landed parts in two Off-Broadway shows, "Kill the One-Eyed Man" and "The Great White Hope". His first major success came when he played "God" in guise of a Puerto Rican steam room attendant in the play "Steambath". Elizondo won an Obie Award for his performance.
In the 1980's, Elizondo befriended Garry Marshall. Marshall was impressed with Elizondo's talent. It was to become a life long friendship which would bring benefits for both. Their first movie together was "Young Doctor's in Love" followed by "The Other Sister". In some of the movies in which Elizondo appeared, he went uncredited and he wouldn't have minded staying that way for the movie "Pretty Woman", however it was Marshall who insisted on crediting him. His role in "Pretty Woman" only lasted 10 minutes, but it was a very successful one and Elizondo received a "Golden Globe" nomination.
Among the roles which Elizondo has played are the following: a Mexican bandit in "Valdez is Coming" (1971), a psychopathic killer in "The Taking of Pelham One Two Three" (1974), a dogged detective in "The Fan" (1981), a middle-class family man in "The Flamingo Kid" (1984), a football coach in "Necessary Roughness" (1991), a Greek coffee shop owner in "Frankie and Johnny" (1991) and as the head of the queens' security service in "The Princess Diaries I and II".