This paper studies a comprehensive framework for reconfigurable intelligent surface (RIS)-assisted integrated communication, sensing, and computation (ICSC) systems with a User-centric focus. The study encompasses two scenarios: the general multi-user equipment (UE) scenario and the simplified single-UE scenario. To satisfy the critical need for time-efficient sensing, we investigate the latency minimization problem, subject to constraints on UEs' transmit power, radar signal-to-interference-plus-noise-ratio (SINR), RIS phase shift, and computation capability. To address the formulated non-convex problem in the multi-UE scenario, we decouple the original problem into two subproblems, where the computational and beamforming settings are optimized alternately. Specifically, for the computational settings, we derive a closed-form solution for the offloading volume and propose a low-complexity algorithm based on the bisection search method to optimize the edge computing resource allocation. Additionally, we employ two equivalent transformations to address the challenge posed by the non-convex sum-of-ratios form in the objective function (OF) of the subproblem related to active and passive beamforming. Several techniques are then combined to address these subproblems. Furthermore, a low-complexity algorithm that offers closed-form solutions is developed for the simplified single UE scenario. Finally, simulation results substantiate the effectiveness of the proposed framework.