For the Mobile Device （MD） with limited computing resources and battery capacity in Mobile Edge Computing （MEC）， its computing capacity can be enhanced and its energy consumption can be reduced through offloading its own computing-intensive applications to the edge server. However， unreasonable task offloading strategy will bring a bad experience for users since it will increase the application completion time and energy consumption. To overcome above challenge， firstly， a multi-objective task offloading problem model with minimizing the application completion time and energy consumption as optimization targets was built in the dynamic MEC network via analyzing the mobility of the mobile device and the sequential dependencies between tasks. Then， a Markov Decision Process （MDP） model， including state space， action space， and reward function， was designed to solve this problem， and a Multi-Objective Task Offloading Algorithm based on Deep Q-Network （MTOA-DQN） was proposed， which uses a trajectory as the smallest unit of the experience buffer to improve the original DQN. The proposed MTOA-DQN outperforms three comparison algorithms including MultiObjective Evolutionary Algorithm based on Decomposition （MOEA/D）， Adaptive DAG （Directed Acyclic Graph） Tasks Scheduling （ADTS） and original DQN in terms of cumulative reward and cost in a number of test scenarios， verifying the effectiveness and reliability of the algorithm.