How to compute the residue of $e^{z-\frac{1}{z}}$ at z=0?

You can use b.gatessucks idea to make it analytically:

f[t_] = Exp[z - 1/z] /. z -> E^(I t) // FullSimplify

then integrate:

1/(2 Pi) Integrate[E^(I t) f[t], {t, 0, 2 Pi}]
(* -BesselJ[1, 2] *)