Since you just want tips, I'll try not to give everything away.
You want to show that $\phi$ can be chosen so that $|\int\phi f|$ is arbitrarily close to $\|f\|_\infty$. To do so, you want to choose $\phi$ to be "concentrated" where $f$ is close to its essential supremum. More explicitly, there is a set $E$ of positive finite measure where $|f|>\|f\|_\infty - \epsilon$, and you can use this to find suitable $\phi$. (Some rescaling and sign adjustments may be necessary.)
If my attempt to give tips is completely opaque, please let me know.