In next-generation mobile communication scenarios, more and more user terminals (UEs) and edge computing servers (ECSs) are connected to the network. To ensure the experience of edge computing services, we designed an unmanned aerial vehicle (UAV)-assisted edge computing network application scenario. In the considered scenario, the UAV acts as a relay node to forward edge computing tasks when the performance of the wireless channel between UEs and ECSs degrades. In order to minimize the average delay of edge computing tasks, we design the optimization problem of joint UE–ECS matching and UAV three-dimensional hovering position deployment. Further, we transform this mixed integer nonlinear programming into a continuous-variable decision process and design the corresponding Proximal Policy Optimization (PPO)-based joint optimization algorithm. Sufficient data pertaining to latency demonstrate that the suggested algorithm can obtain a seamless reward value when the number of training steps hits three million. This verifies the algorithm’s desirable convergence property. Furthermore, the algorithm’s efficacy has been confirmed through simulation in various environments. The experimental findings ascertain that the PPO-based co-optimization algorithm consistently attains a lower average latency rate and a minimum of 8% reduction in comparison to the baseline scenarios.