This work investigates selecting paths for communication flows when deploying a hard real-time application on a chip-multiprocessor system. This chip-multiprocessor system uses a priority-aware real-time network-on-chip interconnect between the processors. Given a mapping of the computation tasks onto the chip-multiprocessor, the problem we address in this work is to discover paths the communication flows take such that hard real-time deadlines of flows are met. Furthermore, we must ensure that deadlines are met even in the presence of direct and indirect interference from other flows sharing network links on the path. To achieve this, our algorithm utilizes a stage-level analysis for real-time communication to determine the impact of a network link being used by a flow, and its effect on other flows sharing the link. The path selection algorithm uses heuristics such as selecting links with least interference, and considering lower-priority flows when dedicating links to paths of higher-priority flows since an optimal one is intractable. The algorithm also considers constraints on the number of virtual channels at each router port in the network. The statistically significant experimental results show an improvement in schedulability by 5% and 12% over existing path selection algorithms such as Minimum Interference Routing and Widest Shortest Path algorithms, respectively. We also present a set-top box case study to further illustrate the benefits of using the proposed algorithm.