Basically, you cannot say anything about value of $f'(0)$. Even for analytic functions (very strong restriction), if we know that $f$ maps $\{z\colon |z|<1\}$ onto (so one-to-one) itself (another very strong restriction) Fractional Linear Transforms show that the only thing you can say that $f'(0)\ne 0$ (and only because one-to-one). On the other hand, if you know also (in addition to all above), that $f(0)=0$, you conclude $|f'(0)|=1$.