This paper investigates outage and throughput performance of non-orthogonal multiple access (NOMA) based underlay cognitive radio (i.e., CR-NOMA) network under partial relay selection (PRS) scheme. Here NOMA is used in the secondary network, where K half duplex, decode-and-forward (DF) relays are available for assisting the secondary base station (SBS) to deliver information to the secondary users (SUs). We derive analytical expressions for the outage probabilities experienced by the SUs under imperfect successive interference cancellation (i-SIC) conditions. The interference threshold constraint of the primary receiver and the maximum transmit power constraint of the secondary nodes are considered for the analysis. To provide more insights on the outage performance, we obtain expressions for the asymptotic outage probabilities of the SUs in the high transmit power region. Next, we derive analytical expressions for the optimal power allocation (OPA) factors at the SBS and at the selected relay that independently maximize the delay limited throughput of the secondary network in CR-NOMA system. We also determine the jointly optimal power allocation factors that maximize the throughput. We use numerical and simulation investigations to establish that the outage probabilities of the SUs reduce significantly while the throughput improves under the considered PRS scheme, compared to random relay selection and fixed relaying strategies. Further, it is established that the proposed OPA factors can significantly reduce the outage probabilities experienced by the SUs and improve the system throughput compared to equal/random power allocation schemes.