Apologies for using a diagram from somewhere else. In the climb the lift acts at right angles to the chord line but the weight acts directly down. The weight can be resolved into two elements, the force that opposes lift, W cos gamma, and the 'drag' element of weight W sin gamma. You can see that W cos gamma (which should show in the diagram as a vector the same length as the lift, but doesn't quite) is less than the weight, so lift is less than weight. Take an extreme case, an F15 going vertical, lift is zero, all the climb comes from the thrust.
Although this answers your question the next bit is to observe that the forces along the axis of the aircraft are Thrust in one direction and Drag and W sin gamma in the other so we can say that for a non-accelerating climb,
T = D + W sin gamma
resolving this,
sin gamma = (T - D)/weight
Which means that the maximum value of sin gamma, in other words the best climb angle, occurs where there is the greatest excess thrust over drag, VMD for a jet, and where the weight is least.