Several of his books won awards and other recognitions in Mexico and abroad. Elizondo was the fourth of six children of Aurora Elizondo of Huinalá and Guadalupe Elizondo of Monterrey proper. Elizondo's father was a worker and union leader at the Fundidora de Fierro y Acero de Monterrey. His mother had served as an arbiter in water disputes in her town. Elizondo's interest in words and stories both fiction and non-fiction began when he was a child. His great grandparents on his mother's side told him many stories of the area, including battles against the indigenous of the area that were still ongoing at the end of the 19th century. This experience formed the basis of his literary development. He began writing stories in his free time since the sixth grade, when he decided he wanted to be a writer. He also began a project of the vocabulary and speech of his region's Spanish, which would eventually become the book "Lexicon del Noreste". Another major influence on his writing was the death of his sister, who was run over by a car when she was seven. The loss shook the family, especially Elizondo, making him more sensible to death, according to his brother Carlos. Although he knew he wanted to be a writer, his family was practical. He attended the Colegio Civil and then went on to study public accounting at the Monterrey Institute of Technology and Higher Studies, graduating in 1975. He did study humanities for a time at UNAM, but he did not finish this degree. Later in life, he would earn a masters in humanities from Universidad de Monterrey and a doctorate in history from the Iberoamericana University. Elizondo retired from his administrative career, but wrote until his death of cancer, which he battled unsuccessfully for two years. Elizondo wrote novels, short stories, plays, works on history and biographies. He first started published short stories about Huinalá and other communities in newspapers when he was seventeen. In the mid-1970s, he was working as an accountant with the Bank of Mexico, when he met writer Juan José Arreola at a café in Mexico City. He presented some of his short stories to the writer, who had a reputation for supporting young talent, who pronounced Elizondo's work “pure silver.” Elizondo wrote for the El Norte and El Porvenir newspapers, wrote biographies of José Vasconcelos and Martín Luis Guzmán along with histories of businesses and institutions.