The study of Service Function Chains (SFCs) placement problem is crucial to support services flexibly and use resources efficiently. Solutions should satisfy various Quality of Service requirements, avoid edge resource congestion, and improve service acceptance ratio (SAR). This work presents a novel approach to address these challenges by solving a multi-objective SFCs placement problem based on the Pointer Network in multi-layer edge and cloud networks. We design a Deep Reinforcement Learning algorithm, called Chebyshev-assisted Actor-Critic SFCs Placement Algorithm, to overcome the limitations of traditional heuristic and evolutionary algorithms. Then, we run this algorithm iteratively with a set of weights to obtain non-dominated fronts, which have much higher hypervolume values than those obtained from other state-of-the-art algorithms. Moreover, running our algorithm individually with selected weights from non-dominated fronts can avoid edge resource congestion and achieve 98% SARs of low-latency services during high-workload periods. Finally, based on both simulation and real testbed experimental results, it is validated that the proposed algorithm fits for pragmatic service deployment while achieving 100% of SARs in the use cases deployed on the testbed.