Fr. Claudio was born in Brooklyn, New York. He is the  oldest son born to the Diaz family. He attended school at the University of Puerto Rico from 1980 to 1984 where he studied Hispanic Studies and Comparative Literature.

He was ordained a priest for the Archdioceses of Chicago on May 20, 2000. In June of 2004 he was appointed by Francis Cardinal George to be the Director of Hispanic Ministry for the Archdiocese of Chicago.

He was the Vice-president of the National Association of Hispanic Priests of the United States also know as ANSH (Asociación Nacional de Sacerdotes Hispanos) from 2009 to 2013. He is part the faculty at the Instituto de Liderazgo Pastoral, located at Mundelein Seminary, where he teaches Liturgy, Homiletics and Pastoral Theology.

He is regularly invited to offer retreats locally, nationally and internationally and continues in his commitment to Hispanic communities, youth and faith formation.