He was the eldest son of Dr Thomas Dampier, who was lower master at Eton College and from 1774 Dean of Durham.

As bishop of Rochester, he proposed an address from the clergy thanking the crown for requiring an undertaking from the ministry not to move in the matter of Catholic emancipation.

The bishopric of Rochester was a poor one, and it was in his case, for the first time for some years past, separated from the deanery of Westminster.

He left a bibliophile's account in Latin, the manuscript of which was extensively used by Thomas Frognall Dibdin in compiling his Aedes Althorpianae.

His library was sold by his half-brother (lawyer Sir Henry) and widow to the Duke of Devonshire at a valuation amounting to nearly £10,000.