Correction in red.

The first step is to substitute the power series for e^z and simplify. Then divide. Have you done this? Where are you stuck? What have you tried?

By the way, after doing the first step it should be obvious that z = 0 is a simple pole. Therefore the series has the form .

The answer you're aiming for is here: Laurent series of 1/(Exp[x] - 1) - Wolfram|Alpha