This paper presents a trajectory optimization problem for rocket landing that accounts for aerodynamic lift and drag, along with its onboard implementation. Considering a rocket landing on Earth, where aerodynamic effects cannot be neglected, the problem formulation incorporates the coupled drag and lift forces parameterized by the angle of attack. In addition, the formulated problem constrains the thrust direction to be aligned with the vehicle attitude, thereby reflecting the typical characteristic of rocket systems with small thrust deflection angle. This formulation preserves additional control authority for rejection of unexpected disturbances. The optimal solution is computed using sequential convex programming, and its algorithm is implemented on an onboard computing platform using an in-house developed custom convex optimization solver to demonstrate its practical feasibility.