Wireless power transfer (WPT) is an effective way to prolong the lifetime of the energy-constraint networks. In this paper, we investigate a wireless powered cooperative non-orthogonal multiple access (WP-CNOMA) system, consisting of a power beacon (PB), an information transmitter (S), multiple relays (R) and two information receiving devices with near device <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$D_{1}$ </tex-math></inline-formula> and far device <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$D_{2}$ </tex-math></inline-formula> . We assume both S and R are energy-constraint and there is no direct link between S and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$D_{2}$ </tex-math></inline-formula> . With the help of PB, S and R can harvest energy from it to restart the communication for WP-CNOMA network. For such a system, low-complexity but effective relay and antenna selection schemes are applied. To characterize the performance, outage probabilities and average throughput are derived for linear and non-linear energy harvesting (EH) models, respectively. Moreover, to maximize the average throughput, invoking the unimodal feature for average throughput with respect to the EH time, we find the optimal EH time via Golden section search method. Simulation results validate the accuracy of analytical results, and reveal the performance gain for our system over the benchmark schemes. Also, it can be seen that the non-linear EH model shows different outage behaviors from the linear one. On the other hand, considering the practical application and to improve the performance, the optimization for a simple WP-CNOMA system with single-antenna PB and single relay is also investigated, in which we aim to maximize the minimum throughput by jointly optimizing EH time and power allocation. A low-complexity analytical method is developed to find the max-min rate. Numerical results show that through optimization, the system performance can be improved significantly.