To straighten the casualty's airway using the head-tilt chin-lift technique, gently place one hand on the casualty's forehead and apply slight pressure to tilt the head back. With the other hand, grasp the chin and lift it upward to open the airway. Ensure that the neck is not excessively extended, especially in cases of potential spinal injury. This position helps to clear any obstruction and allows for better airflow.