In this paper, we investigate a full duplex (FD) multi-user non-orthogonal multiple access (NoMA) communication system based on the optimization of received signal-To-interference-plus-noise ratio (SINR) per unit power. Since the communication system operates in the FD mode, co-channel interference (CCI) and self-interference (SI) dominate the system's performance. Accordingly, to combat the CCI, we adopt a game-Theoretic approach and propose users' clustering algorithms and to suppress the SI, we formulate an optimization problem to maximize the power-normalized SINR (PN-SINR). While the user clustering optimization problem is constrained by: 1) the successive interference cancellation (SIC) constraint and 2) two binary constraints for the allocations of uplink (UL) and downlink (DL) users, the PN-SINR problem is constrained by: 1) total transmit power budget at the base station and UL users; 2) the fundamental condition for the implementation of successive interference cancellation in the NoMA; and 3) the minimum fairness condition for the UL users. The original PN-SINR problem is non-convex and hence is converted into an equivalent subtractive-form problem, after which we propose an iterative algorithm to find the optimal power allocation policy. Properties of all the proposed algorithms are thoroughly investigated and the numerical results are provided. Based on the channel conditions and suppression level of SI and CCI, the superiority of the proposed FD-NoMA system over half-duplex NoMA and FD orthogonal multiple access systems is verified.