He was educated at Trinity College, Oxford, where he took the degree of M.A. on 13 May 1740 and of M.D. on 24 January 1744. He was subsequently admitted a fellow of the College of Physicians, and went into practice in London, but was compelled by bad health to move to the country. His career gained considerable public attention when he appeared as an expert for the prosecution of Mary Blandy for the poisoning of her father Francis Blandy in 1752. He then retired to Reading in Berkshire, where he derived a large income from his profession, until his death in 1790. He devoted his attention particularly to the treatment of insanity, and was one of the physicians called in to see George III when he first showed symptoms of mental aberration.
Dr. Addington was the confidential friend and adviser of Lord Chatham, and took a principal part in negotiating a coalition between that nobleman and Lord Bute. He was unsuccessful in his endeavours, and appears to have made himself enemies, by the account of the matter which he published, under the title of An authentic Account of the Part taken by the late Earl Chatham in a Transaction which passed in the beginning of the year 1778. He was the author of An Essay on the Sea Scurvy, wherein is proposed an easy method of curing that distemper at sea, and of preserving water sweet for any cruise or voyage. Reading, 1753, 8vo. In this work, he describes the disease rather from the accounts of others than from his own observation. In its treatment he recommends depletion, with the employment of seawater as a purgative, and drinks acidulated with muriatic acid. He conceives meat to be injurious, but regards biscuit as food suitable to persons affected with scurvy. He asserts that the addition of an ounce and a half of muriatic acid to a tun of water, will prevent its putrefaction, and preserve it sweet for any length of time.