This paper proposes a novel framework for addressing the challenge of autonomous overtaking and obstacle avoidance, which incorporates the overtaking path planning into Gaussian Process-based model predictive control (GPMPC). Compared with the conventional control strategies, this approach has two main advantages. Firstly, combining Gaussian Process (GP) regression with a nominal model allows for learning from model mismatch and unmodeled dynamics, which enhances a simple model and delivers significantly better results. Due to the approximation for propagating uncertainties, we can furthermore satisfy the constraints and thereby safety of the vehicle is ensured. Secondly, we convert the geometric relationship between the ego vehicle and other obstacle vehicles into the constraints. Without relying on a higherlevel path planner, this approach substantially reduces the computational burden. In addition, we transform the state constraints under the model predictive control (MPC) framework into a soft constraint and incorporate it as relaxed barrier function into the cost function, which makes the optimizer more efficient. Simulation results reveal the usefulness of the proposed approach.