I'm not sure what you are doing will lead anywhere.
Here's what you need to show:
If $f$ is in the unit ball of $C_0$, then there exists two functions $g$ and $h$ in the unit ball of $C_0$ such that $g\ne h$ and $f$ is a nontrivial convex combination of $g$ and $h$.
To show this, you could use the following
Hint: for $f$ in the unit ball of $C_0$, eventually $|f(x)|<1/2$. Using this, find two functions $g$ and $h$ in the unit ball of $C_0$ with $f={1\over2}(h+g)$.
Very informally, you can add a "bump" and subtract the same bump to $f$ over an interval on which the inequality first mentioned holds. $g$ and $h$ will have the forms $f+b$ and $f-b$, where $b$ is the "bump". Do this in such a way that the resulting functions are in the unit ball of $C_0$. Draw a picture here...
I hope that made sense...
