In this paper, we consider a mobile-edge computing system, where an access point assists a mobile device (MD) to execute an application consisting of multiple tasks following a general task call graph. The objective is to jointly determine the offloading decision of each task and the resource allocation under time-varying wireless fading channels and stochastic edge computing capability, so that the energy-time cost (ETC) of the MD is minimized. Solving the problem is particularly hard due to the combinatorial offloading decisions and the strong coupling among task executions under the general dependency model. Conventional numerical optimization methods are inefficient to solve such a problem, especially when the problem size is large. To address the issue, we propose a deep reinforcement learning (DRL) framework based on the actor-critic learning structure. In particular, the actor network utilizes a DNN to learn the optimal mapping from the input states to the binary offloading decision of each task. Meanwhile, by analyzing the structure of the optimal solution, we derive a low-complexity algorithm for the critic network to quickly evaluate the ETC performance of the offloading decisions output by the actor network. With the low-complexity critic network, we can quickly select the best offloading action and subsequently store the state-action pair in an experience replay memory as the training dataset to continuously improve the action generation DNN. To further reduce the complexity, we show that the optimal offloading decision exhibits an one-climb structure, which can be utilized to significantly reduce the search space of action generation. Numerical results show that for various types of task graphs, the proposed algorithm achieves up to $99.1\%$ of the optimal performance while significantly reducing the computational complexity compared to the existing optimization methods.