Copper sulfide concentrates (CSC), processed by smelting to produce copper anodes, may contain precious metals as well as metalloids (e.g., As, Te, and Sb) that partially volatilize during pyrometallurgical processes, presenting operational and environmental challenges. Their selective separation from CSC prior to smelting could improve critical elements supply and bring numerous benefits, which would be reflected in lower penalties imposed by the smelter. Alkaline sulfide leaching (ASL) can selectively extract some of these deleterious elements from CSC. In this paper, ASL was studied as a CSC cleaning method, with the global objective of documenting the leaching behavior of several deleterious and critical element-bearing minerals. Two CSC samples were characterized for chemical and mineralogical composition, as well as particle size distribution. A series of lab-scale ASL tests were performed to evaluate the effect of leaching time, oxygen ingress, number of leaching steps, and regrinding on targeted elements removals. Extractions up to 16%, 74%, and 51% were obtained for As, Te, and Sb, respectively, after 6 h using 75 g/L NaOH, 60 g/L Na2S at 80 °C. Tests showed that oxygen ingress in ASL of the CSC did not improve extraction, while regrinding prior to ASL had a positive effect on the extraction of Sb. Finally, mineralogical analysis shows that both leachable and non-leachable species are present and suggest that the rate of leaching of some species may be limiting the final extraction. This study demonstrates the potential of selective leaching as a means of recovering metals and separating value-added elements, while simultaneously lowering the content of deleterious elements burden in CSC.