Non-orthogonal multiple access (NOMA) serves multiple users by superposing their distinct message signals.The desired message signal is decoded at the receiver by applying successive interference cancellation (SIC). Using the theory of Poisson cluster process (PCP), this paper provides a framework to analyze multi-cell uplink NOMA systems. Specifically, we characterize the rate coverage probability of a NOMA user who is at rank m (in terms of the distance from its serving BS) among all users in a cell and the mean rate coverage probability of all users in a cell. Since the signal-to-interference-plus-noise ratio (SINR) of m-th user relies on efficient SIC, we consider three scenarios, i.e., perfect SIC (in which the signals of m − 1 interferers who are stronger than m-th user are decoded successfully), imperfect SIC (in which the signals of of m − 1 interferers who are stronger than m-th user may or may not be decoded successfully), and imperfect worst case SIC (in which the decoding of the signal of m-th user is always unsuccessful whenever the decoding of its relative m − 1 stronger users is unsuccessful). The worst case SIC assumption provides remarkable simplifications in the mathematical analysis and is found to be highly accurate for scenarios of practical interest. To analyze the rate coverage expressions, we first characterize the Laplace transforms of the intra-cluster interferences in closed-form considering both perfect and imperfect SIC scenarios. In the sequel, we characterize the distribution of the distance of a user at rank m which is shown to be the generalized Beta distribution of first kind and the conditional distribution of the distance of the intracluster interferers which is different for both perfect and imperfect SIC scenarios. The Laplace transform of the inter-cluster interference is then characterized by exploiting distance distributions from geometric probability. The derived expressions are customized to capture the performance of a user at rank m in an equivalent orthogonal multiple access (OMA) system. Finally, numerical results are presented to validate the derived expressions. It is shown that the average rate coverage of a NOMA cluster outperforms its counterpart OMA cluster with higher number of users per cell and higher target rate requirements. A comparison of Poisson Point Process (PPP)-based and PCP-based modeling is conducted which shows that the PPP-based modeling provides optimistic results for the NOMA systems.