Refer to figure.

A steady (continuous) head wind (or tail wind) will not affect your climb rate, only your climb angle.

The angle of climb flight path is the angle the plane is flying with respect to the ground. When flying into a steady headwind the ground speed will be lower at the same airspeed. But the climb rate will be the same. Therefore, if you're climbing at the same rate, but moving forward at a slower rate, the angle will be steeper (increased). It means you will reach a specific altitude in the same time interval, but your ground distance will be affected. In the case of a Headwind, your ground distance will be reduced.

Note: A headwind will not affect the climb angle or rate of climb, but it will affect the angle of climb relative to the ground => it will make the flight path angle steeper.