Hinckley was born in Cove Fort, Utah Territory. He is the uncle of former LDS Church President Gordon B. Hinckley. Alonzo Hinckley married his wife Rose May Robison in 1892 in the Manti Utah Temple. On Sunday, August 11, 1912, during a conference in Millard County, Utah, Hinckley was selected to be president of Deseret Stake, a newly formed stake located in the western half of Millard County and formed from a portion of the Millard Stake.
Hinckley was ordained an apostle on October 11, 1934 after the death of Anthony W. Ivins.
Hinckley died of stomach cancer on December 22, 1936 in Salt Lake City. He was succeeded in the Quorum of the Twelve by Albert E. Bowen.