The use of self-organized micro/nanostructured surfaces formed using femtosecond laser surface processing (FLSP) techniques has become a promising area of research for enhancing surface properties of metals, with many applications including enhancing heat transfer. In this work, we demonstrate advantages of the use of dual-pulse versus single-pulse FLSP techniques to produce self-organized micro/nanostructures on copper. With the dual-pulse technique, the femtosecond pulses out of the laser (spaced 1 ms apart) are split into pulse pairs spaced < 1 ns apart and are focused collinear on the sample surface. Single-pulse FLSP techniques have been widely used to produce self-organized "mound-like" structures on a wide range of metals including a number of stainless steel alloys, aluminum, nickel, titanium, and recently on copper. Due to its high thermal conductivity, copper is used in many critical heat transfer applications and micro/nanostructured copper surfaces are desired to further improve heat transfer characteristics. Using single-pulse (pulses spaced 1 ms apart) FLSP techniques, self-organized microstructure formation on copper requires much higher pulse fluence than is commonly used for producing microstructures on other metals, which results in instabilities during laser processing (non-uniform surfaces), low processing efficiency, and limitations on the control of the types of structures produced. In this paper, we report results that demonstrate that the dual-pulse FLSP technique can be used to produce microstructures on copper more efficiently than using single-pulse FLSP, with better control of the surface structures produced. Cross-sectional subsurface microstructure analysis is also presented for single-pulse versus dual-pulse FLSP functionalized copper surfaces.