A non-orthogonal multiple access (NOMA)-aided joint communication, sensing, and multi-tier computing (JCSMC) framework is proposed. In this framework, a multi-functional base station (BS) carries out target sensing, while providing edge computing services to the nearby users. To enhance the computation efficiency, the multi-tier computing structure is exploited, where the BS can further offload the computation tasks to a powerful Cloud server (CS). The potential benefits of employing NOMA in the proposed JCSMC framework are investigated, which can maximize the computation offloading capacity and suppress the inter-function interference. Based on the proposed framework, the transmit beamformer of the BS and computation resource allocation at the BS and the CS are jointly optimized to maximize the computation rate subject to the communication-computation causality and the sensing quality constraints. Both partial and binary computation offloading modes are considered: 1) For the partial offloading mode, a weighted minimum mean square error based alternating optimization algorithm is proposed to solve the corresponding non-convex optimization problem. It is proved that a KKT optimal solution can be obtained; 2) For the binary offloading mode, the resultant highly-coupled mixed-integer optimization problem is first transformed to an equivalent but more tractable form. Then, the reformulated problem is solved by utilizing the alternating direction method of multipliers approach to obtain a nearly optimal solution. Finally, numerical results verify the effectiveness of the proposed algorithms and the proposed NOMA-aided JCSMC framework