Cohomology rings of $ GL_n(C)$, $SL_n(C)$

For classical groups such as $GL_n(F)$, $SL_n(F)$ for $F = \mathbb{R}, \mathbb{C}$, $SU(n)$, $U(n)$ and $O(n)$, you may find the cohomology ring structure and its proof in M. Mimura and H. Toda, Topology of Lie groups I, translations of mathematical monographs, vol 91.

For $GL_n(\mathbf{C})$ and $SL_n(\mathbf{C})$ we can use the Leray spectral sequences of the mappings to $\mathbf{C}^n\setminus \{ 0\}$ that take a matrix to its last column. For other compact Lie groups (and $\mathbf{Q}$-coefficients) see e.g. A. Borel, Sur la cohomologie des espaces fibrés principaux et des espaces homogènes de groupes de Lie compacts. (French) Ann. of Math. (2) 57, (1953). 115--207. The case of $\mathbf{Z}$ coefficients is much harder, it was settled only recently for simply-connected groups and the answer in general is unknown.

If $G$ is a connected Lie group (or just a connected loop space with finite homology) then $H^*(G,\mathbf{Q})$ is a Hopf algebra. Graded connected Hopf algebras over $\mathbf{Q}$ are always tensor products of exterior algebras in odd degrees with polynomial algebras in even degrees. Since polynomial algebras are infinite, they can't occur. The reference to Hopf is probably H. Hopf, Über die algebraische Anzahl von Fixpunkten, Math. Z. 29 (1929), 493–524. For the classification of Hopf algebras, see also A. Borel, Sur la cohomologie des espaces fibrés principaux et des espaces homogènes de groupes de Lie compacts, Ann. of Math. (2) 57 (1953), 115–207.