This paper focuses on the study of the structure and mechanical properties of CoCrCuxFeNi high-entropy alloys and their adhesion to single diamond crystals. CoCrCuxFeNi alloys were manufactured by the powder metallurgy route, specifically via mechanical alloying of elemental powders, followed by hot pressing. The addition of copper led to the formation of a dual-phase FCC + FCC2 structure. The CoCrCu0.5FeNi alloy exhibited the highest ultimate tensile strength (1080 MPa). Reductions in the ductility of the CoCrCuxFeNi HEAs and the tendency for brittle fracture behavior were observed at high copper concentrations. The equiatomic alloys CoCrFeNi and CoCrCuFeNi demonstrated high adhesion strength to single diamond crystals. The diamond surface at the fracture of the composites having the CoCrFeNi matrix had chromium-rich metal matrix regions, thus indicating that chromium carbide, responsible for adhesion, was formed at the composite-diamond interface. Copper-rich areas were detected on the diamond surface within the composites having the CoCrCuFeNi matrix due to the predominant precipitation of the FCC2 phase at the interfaces or the crack propagation along the FCC/FCC2 interface, resulting in the exposure of the Cu-rich FCC2 phase on the surface.