Born the son of Admiral Sir Edward Codrington and Jane Codrington, Henry Codrington joined the Royal Navy in February 1823. He was initially appointed to the fifth-rate at Portsmouth and then transferred to the fifth-rate HMS Sybille at Deptford in July 1824. Promoted to midshipman, he transferred to the fifth-rate in August 1824 and took park in operations against pirates later in the year, supporting the blockade of Algiers by British forces; he then served in the British squadron off Greece, during the Greek War of Independence. He transferred to the second-rate, flagship of the Commander-in-Chief, Mediterranean Fleet, in October 1826 and was present at the Battle of Navarino in October 1827. During the battle he acted as signal midshipman and was wounded and, following the action during which the Ottoman fleet was destroyed, he was awarded the Russian Order of St. Vladimir, the French Legion of Honour and the Greek Order of the Redeemer for his services. After serving briefly in the third-rate and then in the fifth-rate, Codrington was promoted to lieutenant on 12 June 1829. He was appointed to the first-rate, flagship of the Commander-in-Chief, Portsmouth, in June 1829 and then transferred to the first-rate, flagship of the Commander-in-Chief, The Nore, in August 1829, to the fifth-rate for "particular service" in April 1830 and to the first-rate, flagship of the Commander-in-Chief, Channel Squadron, for an experimental cruise in June 1831. Promoted to commander on 20 October 1831, Codrington became commanding officer of the sloop in the Mediterranean Fleet in June 1834. Promoted to captain on 20 January 1836, he became commanding officer of the sixth-rate in March 1838 and in that capacity undertook a survey of enemy positions prior to the bombardment of Acre in November 1840 during the Egyptian–Ottoman War. For this service he was appointed a Companion of the Order of the Bath. Codrington went on to command the first-rate, his father's flagship as Commander-in-Chief, Portsmouth, in March 1841 and then to command the first-rate, his father's next flagship as Commander-in-Chief, Portsmouth, in October 1841. He became commanding officer of the fifth-rate in the Mediterranean Fleet in October 1846 and provided refuge on board ship for Leopold II, Grand Duke of Tuscany and his family who were fleeing from revolutionary forces in 1848. Codrington became commanding officer of the first-rate in the Baltic Sea in October 1853 and took part in naval operations during the Crimean War. Admiral Sir Charles Napier threatened to court-martial him for failing to achieve the required standards but the Admiralty refused to support this course of action. Promoted to commodore, he was given command of a squadron of gunboats with his broad pennant in the second-rate in February 1856. The Admiralty envisaged that he would lead a mission to attack the naval base at Kronstadt but the War ended with the Treaty of Paris in March 1856 and the mission was abandoned.
In April 1849 Codrington married Helen Jane Webb ; they had two daughters. Following a much publicised divorce in 1864 in which the Admiral accused his first wife of having a close relationship with Emily Faithfull, he married Catherine Aitchison in August 1869; they had one daughter. Ellen Codrington, the younger of the two daughters from Henry Codrington's first marriage, married in 1878 John Roche Dasent, the eldest son of George Webbe Dasent.