Well your contour is a circle of radius 1 centred at (-1, 0).

Your function $\displaystyle \begin{align*} f(z) = \frac{1}{z^4 - 1} = \frac{1}{ \left( z^2 - 1 \right) \left( z^2 + 1 \right) } = \frac{1}{ \left( z - 1 \right) \left( z + 1 \right) \left( z - i \right) \left( z + i \right) } \end{align*}$.

The top and bottom are both holomorphic, so there are simple poles at z = 1, -1, i, -i.

Since only one of these lies inside the contour (z = -1), you can evaluate the integral using the Residue Theorem.