I agree that the question is broad, but here's one sense in which the answer is "all of it". The functor $\rm Grp \to Set$ is represented by the object $\mathbb{Z}$ (the free group on one element); thus we can talk about "elements of a group $G$" in the language of the category $\rm Grp$ by talking about morphisms $\mathbb{Z}\to G$. If we have two such elements $g,h:\mathbb{Z}\to G$, they induce a map $\mathbb{Z}\ast\mathbb{Z}\to G$ from the free group on 2 elements, and there's a map $\mathbb{Z}\to \mathbb{Z}\ast\mathbb{Z}$ such that the composite $\mathbb{Z}\to \mathbb{Z}\ast\mathbb{Z}\to G$ corresponds to the product $g h$. In this way we can extract the entire group -- in the sense of a set with an operation on it -- from an object of $\rm Grp$. (Formally, $\mathbb{Z}$ is a cogroup object in $\rm Grp$, so homming out of it yields a group.) Since certainly all of group theory can be stated as theorems about sets-with-a-group-operation-on-them, we can therefore import it into the (possibly higher-order) language of $\rm Grp$ by rewriting it to refer to maps out of $\mathbb{Z}$ instead of elements. The same can be done for any other algebraic structure like rings, lattices, etc: there's always a co-structure object we can map out of to recover the underlying set with operations. Topology is trickier, but we can detect the points of a space by mapping out of a one-point space, and the opens of a space by mapping into the Sierpinski space.

So far we have to have parameters in our formulas, so our theorems are translated into statements not about $\rm Grp$ alone qua category but about it together with the cogroup object $\mathbb{Z}$. However, in many cases I expect that that object could be uniquely characterized, although I don't know offhand how generally that holds.