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), we provide 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 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.