Dear Student,
Assume the field lines are not perpendicular to the surface of the conductor, then there will be a tangential component of the electric field. Then it would create a potential difference and would accelerate the free electrons resulting in current. The only way a conductor can remain at a constant potential throughout as it must be a equipotential surface, This is only Possible if the electric field lines are perpendicular to all elemental surfaces of the conductor.
Regards