The angle of attack is not increased in the climb. Remember AoA is measured with respect the the relative airflow or relative wind as the Americans call it. Just because the wing is at a greater angle to the horizon in the climb it does not translate that it is at a higher AoA.
The relative airflow, in crude terms, (forgetting about the upwash and downwash) is opposite to the direction of travel. During a climb the aircraft is moving forward and upward at the same time.
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. It is thrust that makes the plane climb, the more excess thrust you have over and above that required to pull the aircraft thru the air the greater your rate of climb.
Once thrust exceeds weight you are able to climb vertically, in this instance the wings are producing no lift that supports the aircrafts weight.