As Willie already said, you should look at the action of your group on the the extreme rays of C. Look at the kernel of this action, K. The group GL(C)/K is then finite (and bounds on its order can be derived from the fact that it will be a permutation action, that is realised in a subspace of certain dimension...)
Regarding K, $K$, there will be two cases:
1) the number a partition of rays equals the dimension d set of the space spanned by C, and then K could scale each extreme ray independently
2) this number is bigger than d; then K can only scale all rays into components $I_1,\dots,I_k$, so that each basis of them simultaneously, i.e. K acts on the linear span of C by (positive) scalar multiplications.
The case 1) is kind $C$ consisting of trivial extreme rays will have $\dim (V_{I_j})$ elements from $V_{I_j}$, the linear span of $I_j$. Then $K$ will induce the multiplication by positive scalars action on each $V_{I_j}$, and GL(C)/K is quite obvious)will be the direct product of these actions.
(Thanks to David Speyer for pointing the error in the original description of $K$).
