Clement Archer was born in County Wexford on 21 December 1748.
He settled in Dublin in 1774 and was an original member of the Surgeons' Society.
In 1785, he, together with surgeons Bolger, Lindsay, Costelloe, Hartigan, and Graydon, and Drs.
Brereton, Percival, Dickson, Kennedy, Bell, and Boyton, founded the Dublin General Dispensary in the old Post Office yard, Temple Bar, the treasurer being Sir William Newcomen.
Archer was one of 49 physicians and chirurgeons who declared their public support for the construction of a Publick Bath in Dublin in May 1771 and named Achmet Borumborad as a well qualified individual for carrying such a scheme into existence.