This paper presents a local trajectory planning method based on the Rapidly-exploring Random Tree (RRT) algorithm using Dubins curves for autonomous racing vehicles. The purpose of the investigated method is the real-time computation of a trajectory that could be feasible in autonomous driving. The vehicle is considered as a three Degree-of-Freedom bicycle model and a Model Predictive Control (MPC) algorithm is implemented to control the lateral and longitudinal vehicle dynamics. The trajectory planning algorithm exploits a perception pipeline using a LiDAR sensor that is mounted onto the front wing of the racing vehicle. The MPC computes the acceleration/ deceleration command and the front wheel steering angle to follow the predicted trajectory. The trajectory and control algorithms are tested on real data acquisition performed on-board the vehicle. For validation purposes, the vehicle is driven autonomously during different maneuvers performed in the racing environment that is structured with traffic cones. The feasibility of the algorithm is evaluated in terms of error with respect to the planned trajectory, tracking velocity and maximum longitudinal acceleration. The effectiveness of the method is also evaluated with respect to command signals for the steering and acceleration actuators featured by the retained racing vehicle. The results demonstrate that the trajectory is well-tracked and the signals are compatible with the actuator constraints.