In the finite case at least, the answer is yes. It is a tensor category, and one fundamental result is that the category of representations of G is equivalent, as tensor categories, to the category of $\mathbb{C}G$-modules, where $\mathbb{C}G$ is the group algebra. The same holds if you just look at the finite dimensional representations, and finite dimensional modules if I remember correctly.

As for considering it as a ring, the answer is also yes. It has a basis the irreducible representations, and the multiplication is the tensor product, with the trivial representation as 1.

In the finite case at least, the answer is yes. It is a tensor category, and one fundamental result is that the category of representations of G is equivalent, as tensor categories, to the category of $\mathbb{C}G$-modules, where $\mathbb{C}G$ is the group algebra. The same holds if you just look at the finite dimensional representations, and finite dimensional modules if I remember correctly.

As for considering it as a ring, the answer is also yes, if you allow formal combinations of them (so called virtual representations) (thanks for pointing that out Jose and Qiaochu!). It has a basis the irreducible representations, and the multiplication is the tensor product, with the trivial representation as 1.