For future wireless communication networks to be highly efficient with both spectrum and energy, reconfigurable intelligent surface (RIS) has emerged as a practical technology. In this paper, we investigate an RIS-aided cooperative non-orthogonal multiple access (C-NOMA) system serving two power-domain users, where a near user as a decode-and-forward (DF) half-duplex (HD) relay performs device-to-device (D2D) communications for a far user. RIS is configured in time division multiple access (TDMA), which is on the one hand used to support the communication between BS and near user in the direct transmission phase, and on the other hand, for enhancing D2D communications between the two users. In an effort of evaluating the network performance, we derive closed-form expressions of achievable rates, outage probabilities (OPs), and the corresponding asymptotic OPs. The numerical results demonstrate the achievable performance improvements of the proposed cooperative RIS-NOMA system in comparison to that of the orthogonal multiple access (OMA) counterpart. In addition, the proposed cooperative RIS-NOMA system achieves various levels of diversity gains by incorporating different numbers of RIS elements.