This paper presents a physics-informed neural network (PINN)-based solution framework that predicts the thermal history during a multi-layer Directed Energy Deposition (DED) process. The meshless nature and the readily available derivative information of PINN solution opens up new opportunities for modelling the thermally induced distortion in metal Additive Manufacturing (AM). The proposed framework incorporates simple yet effective strategies that enable PINN to overcome the usual shortfall of neural networks (NNs) in dealing with discontinuities. It is a critical step for applying PINN to the multi-layer problem which intrinsically contains discontinuities due to the layer-by-layer nature of DED and other metal AM processes. The accuracy of the proposed framework is validated via a benchmark test against ANSYS simulation. Leveraging the possibility of initialisation with prior knowledge, PINN is also demonstrating potential computational time-savings, especially for larger parts. Furthermore, remarks on strategies to improve ease of training and prediction accuracy by PINN for the particular use casein DED temperature history prediction have been made. The proposed framework sets the foundation for the subsequent exploration of applying scientific machine learning (SciML) techniques to real-life engineering applications.