Hamilton chose a career in the Royal Navy and served in Guinea and the West Indies from 1737 to 1740.
He was promoted to captain the next year and served throughout the War of the Austrian Succession, mostly in escorting convoys.
His portrait by Joshua Reynolds in 1746 brought the artist his initial fame.
He remained on active service for a few years after the close of the war in 1748, but went on half-pay from 1751 to 1755.
He was called back into service as commander of HMS Lancaster, and accidentally drowned while in Portsmouth harbour.