Tapia was born in San Juan, Puerto Rico. There he received his primary education under the guidance of the renowned educator Rafael Cordero while studying his primary years in a school in Luna Street (Calle Luna) in Old San Juan, who was an early inspiration in his life. There he met his life-long friend Jose Julian Acosta.

Tapia worked for the State Department. A Spanish Army artillery officer challenged him to a duel when Tapia refused to yield the sidewalk to him, a challenge which he accepted; as a result, Tapia suffered a non-lethal injury in the arm and subsequently exiled to Spain, where he remained with his father for a period of time.

While in Spain he completed his studies in literature in Madrid (1850—1852) and joined the Society of the Recollection of Historical Documents Relating to Puerto Rico (Sociedad Recolectora de Documentos Históricos de San Juan Bautista de Puerto Rico). Tapia filed and organized important 16th and 17th century documents of Puerto Rico. He published his first important work called The Historical Library of Puerto Rico (Biblioteca Histórica de Puerto Rico) relating to those documents and his findings on them.

In 1852, the Spanish appointed governor of Puerto Rico, Fernando de Norzagaray, pardoned Tapia and returned to Puerto Rico, establishing his residence in the City of Ponce on the southern coast of the island. He was a member of the Progressive Action Political Party. Among his cultural positions was his membership in the Puerto Rican Intellectual Protective Society. He was also the director of the Youth Museum in Ponce and the founder and first president of the Ateneo Puertorriqueño.

Alejandro Tapia y Rivera died in the city of San Juan, Puerto Rico on July 19, 1882, while giving a conference at the Ateneo Puertorriqueño.

Among Tapia's many awards and honours were:

• The Medal of the Royal Knight (Medalla de la Orden del Caballero Real) and
• The Royal and Distinguished Order of Carlos III from Spain.

His memory has been honoured in Puerto Rico. There are several schools and avenues named after Tapia. The Teatro Tapia, the premier drama stage, in old San Juan is also named after him. In July 2008, Roberto Ramos Perea, general director of the Archivo Nacional de Teatro y Cine del Ateneo Puertorriqueño, amn archive dedicated to conserve documents and memorabilia from the theatre industry in Puerto Rico, announced that a documentary covering Tapia y Rivera's life is planned. The filmation is part of a series of events honoring his life, titled Jornadas en Honor y Memoria de Alejandro Tapia y Rivera.