In this paper, we study the application of cooperative non-orthogonal multiple access (NOMA) in power line communication (PLC) where two users exchange information using a two-way relay (TWR). Under the assumption of perfect and imperfect successive interference cancellation (SIC), we analyze key performance metrics for PLC log-normal channels impaired by impulsive noise. In particular, for delay-limited (DL) and delay-tolerant (DT) modes, approximate closed-form expressions are derived for the user outage probability and user ergodic rate. The analytic results are verified via Monte Carlo simulations. Compared with the conventional orthogonal multiple access (OMA) scheme, the simulation results show that our proposed NOMA TWR enhances the user outage probability and the user ergodic rate performance in the low signal-to-noise ratio (SNR) region. However, this gain diminishes as the imperfect SIC error increases and is non-existent beyond a certain level of imperfect SIC. Furthermore, the system performance is maximized by optimizing the power allocation coefficients where the system throughput and ergodic sum-rate of the NOMA TWR network in DL and DT operation, respectively, are shown to approach a ceiling in the presence of imperfect SIC. The energy efficiency (EE) of the proposed system is also analyzed for DL and DT modes.