Mobile edge computing (MEC) networks bring computing and storage capabilities closer to edge devices, which reduces latency and improves network performance. However, to further reduce transmission and computation costs while satisfying user-perceived quality of experience, a joint optimization in computing, pushing, and caching is needed. In this paper, we formulate the joint-design problem in MEC networks as an infinite-horizon discounted-cost Markov decision process and solve it using a deep reinforcement learning (DRL)-based framework that enables the dynamic orchestration of computing, pushing, and caching. Through the deep networks embedded in the DRL structure, our framework can implicitly predict user future requests and push or cache the appropriate content to effectively enhance system performance. One issue we encountered when considering three functions collectively is the curse of dimensionality for the action space. To address it, we relaxed the discrete action space into a continuous space and then adopted soft actor-critic learning to solve the optimization problem, followed by utilizing a vector quantization method to obtain the desired discrete action. Additionally, an action correction method was proposed to compress the action space further and accelerate the convergence. Our simulations under the setting of a general single-user, single-server MEC network with dynamic transmission link quality demonstrate that the proposed framework effectively decreases transmission bandwidth and computing cost by proactively pushing data on future demand to users and jointly optimizing the three functions. We also conduct extensive parameter tuning analysis, which shows that our approach outperforms the baselines under various parameter settings.