Born in Las Anonas, near San José, Costa Rica, Zeledón was the scion of a distinguished family whose intellectual gifts and love of learning were common traits. His father, don Manuel Zeledón, was governor of the district of San José for thirty years, and a man of great integrity. From earliest childhood José was interested in birds, and began serious study as a naturalist under the tutelage of Alexander von Frantzius, who arrived in Costa Rica in 1854. Enroute to Germany in 1868, Dr. Frantzius took Mr. Zeledón to Washington, D.C., where he was introduced to Spencer Baird and installed as an unpaid assistant in the Smithsonian Institution. Here he had the opportunity to study the bird collections, and acquaint himself with an already impressive library of ornithological literature. Zeledón stayed at the Institution to study with other naturalists, meeting among others Dr. Robert Ridgway with whom he began a lifelong friendship. In 1871 he returned to Costa Rica as part of a scientific expedition organized by Dr. William Gabb, to explore the wilderness of Talamanca, where he made a large and important collection of birds.

Zeledón frequently exchanged natural history and archaeological artifacts from Costa Rican collections with the Smithsonian's National Museum. One of José Zeledón's first significant publications, "Catalogue of the Birds of Costa Rica" was published by the Smithsonian's United States National Museum in 1885. In 1886, Zeledón presented a magnificent collection of birds comprising 1500 examples at the first Costa Rican Exposition. In this collection Zeledón had nearly every Costa Rican species, which at that time numbered 700.

He was one of the founders of the National Museum in Costa Rica, and was one of it's earliest and chief benefactors. His principal interest was ornithology, with scores of his discoveries bearing his name. Among these were a number of type specimens now in the collections of the National Museum. He died suddenly in Turin, Italy, in 1923 at the age of 77.