In this work, we propose a cross-layer design strategy based on a joint successive interference cancellation (SIC) detection technique and a multi-relay selection algorithm for the uplink of cooperative direct-sequence code-division multiple access (DS-CDMA) systems. We devise a low-cost greedy list-based SIC (GL-SIC) strategy with RAKE receivers as the front-end that can approach the maximum likelihood detector performance. %Unlike prior art, the proposed GL-SIC algorithm %exploits the Euclidean distance between users of interest, multiple %ordering and their constellation points to build an effective list %of detection candidates. We also present a low-complexity multi-relay selection algorithm based on greedy techniques that can approach the performance of an exhaustive search. %A cross-layer %design strategy that brings together the proposed GL-SIC algorithm %and the greedy relay selection is then developed. Simulations show an excellent bit error rate performance of the proposed detection and relay selection algorithms as compared to existing techniques.