The max norm $|. |_{\infty}$ is generating a metrics 
I want to prove that the max norm $\|\cdot \|_{\infty}$ generates in the
  space of continuos functions $C([a,b])$ a metrics.

Well, I am not sure how to properly make it. I just know that if I have two functions $f$ and $g$ then for all $x \in [a,b]$ a I can find the distance between those two functions such that $\|f - g\|$. For the $x \in [a,b]$ I can find out the biggest possible distance between $f(x)$ and $g(x)$ by $max_{x \in [a,b]}|f(x) - g(x)|$. Is this the metrics or do I have to do more or something else?

You are, I think, trying to say:
The norm $\|\cdot\|_\infty$ generates a metric on $C([a,b])$ by $$d(f,g) = \|f-g\| = \max_{x \in [a,b]} \{| f(x) -g(x) |\}$$
This is true, as any norm $\|\cdot\|$ on a linear space defines a metric in this way (norm of the difference).
We use $\|-f\| = \|f\|$ for symmetry, $\|f\| = 0 \implies f=0$ for the non-pseudometricity, and $\|f+g\| \le \|f\| + \|g\|$ for the triangle inequality of that metric.
