Concave production function implies convex cost function
Given the fixed input price $w$, the cost function can be written as
$$
C(q)=f^{-1}(q)\times w
$$
where $f^{-1}$ is the inverse of the production function $f$. From the discussion here, one can conclude that the inverse of a concave strictly increasing function is convex. Thus, $C(q)$ is convex as well. 

Going back to your approach, you might like to have this clearly stated. Let $q''=\alpha q + (1-\alpha)q'$, $f(z)=q$, $f(z')=q'$, and $f(z'')=q''$ Then 
$$
\begin{align}
f(z'')&amp;=&amp;q''\\
&amp;=&amp;(\alpha q + (1-\alpha)q')\\
&amp;=&amp;\alpha f(z)+(1-\alpha) f(z')\\
&amp;\leq&amp;f(\alpha z +(1-\alpha z'))
\end{align}
$$
Then $f(z'')\leq f(\alpha z +(1-\alpha z'))$ implying that $\alpha z +(1-\alpha z')\geq z''$ since $f$ is (strictly) increasing. Hence, 
$$
z''w=C(q'')\leq \alpha C(q)+ (1-\alpha) C(q')
$$