Using the chain rule we have the following expressions for say one of the weight parameters (it is easy to generalize to the other weight parameters)
$$ \frac{\partial C}{\partial w_{11}^{(1)}}=\frac{\partial C}{\partial a^{(2)}}\frac{\partial a^{(2)}}{\partial z^{(2)}} \frac{\partial z^{(2)}}{\partial z_1^{(1)}}\frac{\partial z_1^{(1)}}{\partial w_{11}^{(1)}}= \delta^{(2)}\frac{\partial z^{(2)}}{\partial z_1^{(1)}}\frac{\partial z_1^{(1)}}{\partial w_{11}^{(1)}}, $$which, noting that
$$ z^{(2)} =w_1^{(2)}a_1^{(1)}+w_2^{(2)}a_2^{(1)}+b^{(2)}, $$allows us to rewrite
$$ \frac{\partial z^{(2)}}{\partial z_1^{(1)}}\frac{\partial z_1^{(1)}}{\partial w_{11}^{(1)}}=w_1^{(2)}\frac{\partial a_1^{(1)}}{\partial z_1^{(1)}}a_1^{(1)}. $$