The method Mariano discusses in his answer is absolutely the way that mathematicians compute fundamental groups (and also higher homotopy groups) of Lie groups. Here I just want to mention how his first step applies in a more general context.
1) Concerning $\operatorname{GL}_n(\mathbb{C})$: the unitary group $U(n)$ is a maximal compact subgroup of $\operatorname{GL}_n(\mathbb{C})$, and moreover any maximal compact subgroup is conjugate to $U(n)$. (This can be seen by considering Hermitian forms, c.f. e.g. Section 1 of http://math.uga.edu/~pete/8410Chapter9.pdf.) Moreover, the Gram-Schmidt process gives a deformation retraction from $\operatorname{GL}_n(\mathbb{C})$ to $U(n)$, hence these two spaces are homotopy equivalent. And in fact even more is true: there exists a finite-dimensional Euclidean space $E$ such that $\operatorname{GL}_n(\mathbb{C})$ is homeomorphic to $U(n) \times E$: this is the QR decomposition. Moreover:
2) Everything in 1) goes over verbatim for $\operatorname{GL}_n(\mathbb{R})$ with the unitary group $U(n)$ replaced by the orthogonal group $O(n)$.
3) For any reductive group $G$ over $\mathbb{R}$ or $\mathbb{C}$, there exists a maximal compact subgroup $K$, any two such are conjugate in $G$, and $G$ is homeomorphic to the product of $K$ with a finite-dimensional Euclidean space. This last fact is a consequence of the Iwasawa decomposition, a far-reaching generalization of the QR-decomposition.