Refer to figure.
In a steady climb with a climb angle of γ, the aircraft is moving at a steady speed. Therefore, there is no net force on the aircraft - the forces acting up = the forces acting down, and the forces acting forwards = the forces acting backwards. We can draw out a diagram of the forces acting on a climbing aeroplane, and using trigonometry, can derive the following equations for a steady climb:

L = W Cos(γ)

T = D + W Sin(γ)

For any positive climb angle γ, we can see that lift is less than weight. This might initially seem counterintuitive as the aircraft is climbing - but it is because the force which climbs the aircraft is not an excess of lift, but excess thrust. Excess thrust is when there is more thrust than the amount of drag (and is equal to T - D). This is how a rocket can climb without lift, it is engine thrust that accelerates it upwards to climb. 

This also means that for a steady climb, thrust is always greater than drag.