Additive manufacturing of copper gained a high research interest in recent years. Pure copper is challenging to manufacture in powder bed fusion with common infrared laser systems, due to its high reflectivity. Additionally, laser powder bed fusion (LPBF) processes are limited regarding the achievable surface qualities and microstructures. Postprocessing techniques are therefore utilized to adjust surface and microstructure properties. In this article, a 532 nm laser is used to process copper via LPBF and the possibilities to alter surface roughness and microstructure are explored. To change the surface characteristics electrolytic coatings are applied as well as shot peening and vibratory grinding. The possibilities to adjust the microstructure and mechanical properties with heat treatment are explored. Changes in mechanical behavior are tested on lattice structures with compression testing to provide an insight into their mechanical properties, as they offer a high potential for adjusting mechanical and functional properties and can feasibly manufactured via LPBF. As a result, all tested surface treatments are suitable to reduce the roughness of copper after LPBF. Heat treatments show a significant influence on the microstructure regarding grain morphology and on the mechanical behavior during compression testing.