In this paper, we analyze the performance of a full-duplex (FD) cooperative non-orthogonal multiple access (NOMA) model in multi-relay multiuser networks. In contrast to conventional cooperative networks, relays in the considered network have no embedded energy supply, but they can harvest energy from the radio-frequency signals broadcasted by the source to facilitate the cooperative NOMA transmission. Based on this structure, we propose a novel relay selection scheme in Nakagami-m fading channels by considering both the channel state information (CSI) and the battery statuses of relays. Assuming each relay has finite energy storage for the energy accumulation, we adpot a finite Markov chain to capture the evolution of relay batteries and simplify the performance analysis by making some reasonable assumptions. General closed-form expressions of the outage probability and the ergodic sum-rate are derived. All the theoretical results are validated by Monte-Carlo simulations. The impacts of various system parameters, such as the number of relays and the self-interference (SI) at the involved relay, on the performance are extensively investigated. It is shown that the FD relaying achieves better performance in the moderate and high SNR regions and the usage of NOMA with FD relaying outperforms the conventional orthogonal multiple access (OMA) network in terms of terminal fairness.