Hastings Henry, as he then was, was born in March 1808, the son of John Joseph Henry, of Straffan, and Lady Emily Elizabeth FitzGerald, daughter of the second duke of Leinster. On 20 August 1823, the 15-year-old Henry entered the Royal Navy as a first-class volunteer aboard Sybille, under Captain Samuel Pechell. While serving aboard Sybille, then in the Mediterranean, Henry was present for a hard-fought engagement with pirates off Candia in 1826. He subsequently served as a midshipman and mate aboard Columbine, Undaunted, and St Vincent, in home waters.
Promoted lieutenant on 18 December 1830, he went aboard Asia, flagship of Sir William Parker. Asia was then at Lisbon, protecting British interests during the Portuguese Civil War. In 1834, Lieutenant Henry joined Rattlesnake, under Captain Hobson, on the East Indies station. He was promoted commander on 28 June 1838. Henry returned to the home and Mediterranean stations from 1840 and 1843, serving aboard Styx and Devastation and as acting captain of Queen and Aigle. His promotion to post captain came on 23 September 1843.
In 1853, Captain Yelverton was given command of Arrogant, a steam screw frigate, and took part in the Baltic campaigns. On 19 May 1854, Arrogant and Hecla cut out a Russian barque beneath the batteries at Eknas, Finland. Throughout much on 1855, Yelverton operated independently destroying Russian installations along the Finnish coast, and was created a CB for his efforts. In 1856, he took command of the battleship Brunswick and a gunboat flotilla for further operations, but the Crimean War ended before he saw service in that command. Yelverton was appointed comptroller-general of the coastguard on 3 August 1859, and held the post until April 1863.
Again in command of the Channel Fleet from July to October 1870, he then became Commander-in-Chief, Mediterranean Fleet, until January 1874. During this period, he took part in the suppression of the Cartagena revolt, which at the time commanded much of the Spanish navy. On 29 May 1875, he was made GCB.