Da Costa was born in Lobito, then a lively coastal town in the Benguela province of Angola, to a white Portuguese father and a mother from one of the local tribes. In early 1976 Da Costa arrived at Buffalo Base in the Western Caprivi with a group of ex-FNLA soldiers, who would later become the formidable fighting force known as 32 Battalion.

Da Costa's first opportunity to do specialised work came with the formation of the reconnaissance wing of 32 Battalion. He joined up, and by the end of 1977 had moved to Omauni, the recce wing's operational base. By 1981 he had made up his mind to join Special Forces.