He married Jane, daughter and heiress of Robert Selby of Bonnington, near Edinburgh in 1783, by whom he had issue: Norborne Charles (1785–1826) who joined the navy but was court martialed for insubordination; Charles Robert (1788–1801) who died at sea aged 13; Elizabeth (1790-, Jane (1794–1815) who died in Portugal aged 21, and is buried at the English Cemetery, Lisbon; and Henry (1796–1868).

Thompson's first service at sea was on a merchantman, but he soon joined the Royal Navy on HMS Nassau in 1755, just before the outbreak of the Seven Years' War.

He took the later ship back to England in December 1771 and, though his acting captaincy was not confirmed by the admiralty, they did on 7 March 1772 promote him to full captain, commanding HMS Chatham.

[1] He returned to England in 1774, and then went back to the West Indies in command of HMS Boreas in early 1776 (capturing the 20-gun French ship Le Compas).

Sir John Laforey was appointed commissioner of the shipyard at Antigua in 1780, but Thompson refused to recognize this authority, leading to a long feud.

When the War of the First Coalition broke out in 1793 he was put in command of HMS Vengeance, as part of Sir John Jervis and Lieutenant-General Sir Charles Grey's expedition against France's West Indian possessions, participating in the capture of Martinique (directing the boat attacks on Fort Royal) and Guadeloupe as a commodore.

Thompson was next put in command of a detached squadron as part of the British blockade of Brest, before being transferred to HMS Britannia, in which he served in the Mediterranean.