The AoA in the climb is about the same as it would be for the same speed in straight and level with an allowance for that proportion of weight that is being supported by part of the vertical component of thrust.
In a climb the thrust is angled upwards compared to the straight and level position.
Not only in climb.
Consider plane in a steady level flight. At some AoA, suppose that the thrust is exactly horizontal.
Since the flight is level, drag is exactly horizontal (and equal to thrust) while lift is exactly vertical (and equal to weight).
Now, pitch the plane up, increasing Cl, and slow it down, decreasing Vsquared, so that the plane is again in steady level flight.
Since the flight is again level, the drag must needs be exactly horizontal. But the thrust is now angled upwards, since we just pitched the whole plane with its engines. Therefore some of the weight of the plane is supported by the vertical component of the thrust. Lift must be smaller than weight - in level steady flight.
Indeed, as you pitch up to steady hover, both lift and drag must go to zero (because airspeed is zero) and thrust must be exactly equal to lift.
But when you go to steady vertical climb, lift and drag are no longer required to be zero because there is nonzero airspeed. Drag is opposed to airspeed, which means directly down. Lift must be at a right angle to airspeed, so horizontal. Thrust no longer needs to be directly vertical when there is lift present.