Integrated sensing and communication (ISAC) capability is envisioned as one key feature for future cellular networks. Classical half-duplex (HD) radar sensing is conducted in a "first-emit-then-listen" manner. One challenge to realize HD ISAC lies in the discrepancy of the two systems' time scheduling for transmitting and receiving. This difficulty can be overcome by full-duplex (FD) transceivers. Besides, ISAC generally has to comprise its communication rate due to realizing sensing functionality. This loss can be compensated by the emerging reconfigurable intelligent surface (RIS) technology. This paper considers the joint design of beamforming, power allocation and signal processing in a FD uplink communication system aided by RIS, which is a highly nonconvex problem. To resolve this challenge, via leveraging the cutting-the-edge majorization-minimization (MM) and penalty-dual-decomposition (PDD) methods, we develop an iterative solution that optimizes all variables via using convex optimization techniques. Besides, by wisely exploiting alternative direction method of multipliers (ADMM) and optimality analysis, we further develop a low complexity solution that updates all variables analytically and runs highly efficiently. Numerical results are provided to verify the effectiveness and efficiency of our proposed algorithms and demonstrate the significant performance boosting by employing RIS in the FD ISAC system.