Model Predictive Controller (MPC) is a capable technique for designing Path Tracking Controller (PTC) of Autonomous Vehicles (AVs). The performance of MPC can be significantly enhanced by adopting a high-fidelity and accurate vehicle model. This model should be capable of capturing the full dynamics of the vehicle, including nonlinearities and uncertainties, without imposing a high computational cost for MPC. A data-driven approach realised by learning vehicle dynamics using vehicle operation data can offer a promising solution by providing a suitable trade-off between accurate state predictions and the computational cost for MPC. This work proposes a framework for designing an MPC with a Neural Network (NN)-based learned dynamic model of the vehicle using the plethora of data available from modern vehicle systems. The objective is to integrate an NN-based model with higher accuracy than the conventional vehicle models for the required prediction horizon into MPC for improved tracking performances. The proposed NN-based model is highly capable of approximating latent system states, which are difficult to estimate, and provides more accurate predictions in the presence of parametric uncertainties. The results in various road conditions show that the proposed approach outperforms the MPCs with conventional vehicle models.