Your question is too general because you don't specify $g(z)$ and $C$. Something can be done for more concrete cases e.g. $g(z):=z^n,\, n \in \mathbb{N}, C:=\{z:|z|=2\}$ with Maple 2019.1:

int(I*eval(exp(1/z) * exp(2/(z - 1)), z = 2*exp(t*I))*2*exp(t*I), t = 0 .. 2*Pi,numeric);

${ 3.828525440\times 10^{-15}}+ 40.84070450\,i $

identify(40.84070450*I);

$13\,i\pi $

I think the same can be done with Mathematica.

Your question is too general because you don't specify $g(z)$ and $C$. Something can be done for more concrete cases e.g. $g(z):=z^n,\, n \in \mathbb{N}, C:=\{z:|z|=2\}$ with Maple 2019.1:

int(I*eval(exp(1/z) * exp(2/(z - 1)), z = 2*exp(t*I))*2*exp(t*I), t = 0 .. 2*Pi,numeric);

$ { 1.788426568\times 10^{-14}}+ 18.84955592\,i$

identify(18.84955592*I);

$6\,i\pi $

I think the same can be done with Mathematica.

Addition. Making use of the residue at infinity and Maple, one obtains a simple symbolic result

residue(exp(1/z)*exp(2/(z - 1)), z = infinity + infinity*I) 

$-3\pi i$

Therefore, the above integral equals $6\pi i$. Under some conditions on $g(z)$ this should work in the general case too.

Your question is too general because you don't specify $g(z)$ and $C$. Something can be done for more concrete cases e.g. $g(z):=z^n,\, n \in \mathbb{N}, C:=\{z:|z|=2\}$ with Maple 2019.1:

int(2*eval(exp(1/z) + exp(2/(z - 1)), z = 2*exp(t*I))*exp(t*I), t = 0 .. 2*Pi);

$6\,\pi$

int(2*eval(z^4*(exp(1/z) + exp(2/(z - 1))), z = 2*exp(t*I))*exp(t*I), t = 0 .. 2*Pi);

${\frac {2513\,\pi}{60}}$

Unfortunately, a more general case

int(2*eval(z^n*(exp(1/z) + exp(1/(z-1))),z=2*exp(t*I))*exp(t*I), t = 0 .. 2*Pi)assuming n::posint;

fails.

Your question is too general because you don't specify $g(z)$ and $C$. Something can be done for more concrete cases e.g. $g(z):=z^n,\, n \in \mathbb{N}, C:=\{z:|z|=2\}$ with Maple 2019.1:

int(I*eval(exp(1/z) * exp(2/(z - 1)), z = 2*exp(t*I))*2*exp(t*I), t = 0 .. 2*Pi,numeric);

${ 3.828525440\times 10^{-15}}+ 40.84070450\,i $

identify(40.84070450*I);

$13\,i\pi $

I think the same can be done with Mathematica.