See H. Cole, The Murats (1972).
Joachim Joseph was born at Bordertown, New Jersey, as the eldest son of Prince Napoleon Lucien Charles Murat, 2nd Prince of Pontecorvo and 3rd Prince Murat, the second son of Joachim Murat, King of Naples, who married Napoleone's sister, Caroline Bonaparte. His mother was Caroline Georgina Fraser (Charleston, South Carolina, April 13, 1810 - Paris, February 10, 1879), daughter of Thomas Fraser. He had other four siblings.
In 1834 he married at the Tuileries Palace Malcy Louise Caroline Berthier de Wagram (Paris,June 22,1832 - Paris, May 17, 1884), daughter of Napoléon Berthier de Wagram, 2nd Duc de Wagram (September 10, 1810 - February 10, 1887, son of Marshal Berthier) and wife Zénaïde Françoise Clary (Paris, November 25, 1812 - Paris, April 27, 1884, niece of Désirée Clary and Julie Clary), and had issue (two daughters and one son, the 5th Prince Murat), and secondly in Paris, November 7, 1894 Lady Lydia Hervey (Kemptown, Sussex, August 15, 1841 - Château Chaâlis, September 25, 1901), without issue.
After the fall of Napoleon III he retired to a private life but was able to maintain the title of General and Prince. He spent the rest of his life at his family's castle in Chambly where he died in 1901.