Can every group be represented by a group of matrices? 
Can every group be represented by a group of matrices?

Or are there any counterexamples? Is it possible to prove this from the group axioms?

It is not true that every group can be represented by a group of finite-dimensional matrices (say over $\mathbb{C}$). The groups that can are called linear. There are many examples of non-linear groups; here is a relatively simple one.  
Claim: The group $(\mathbb{Z}/2\mathbb{Z})^{\infty}$ is not linear.
Proof. Suppose to the contrary that there exists a faithful representation $(\mathbb{Z}/2\mathbb{Z})^{\infty} \to \text{GL}_n(\mathbb{C})$ for some $n$. In particular, for arbitrarily large $m$, there exists a faithful representation $(\mathbb{Z}/2\mathbb{Z})^m \to \text{GL}_n(\mathbb{C})$. We can conjugate this to a representation into $U(n)$ and then simultaneously diagonalize to obtain a representation into $\mathbb{T}^n$. But the subgroup of elements of $\mathbb{T}^n$ of order $2$ is $(\mathbb{Z}/2\mathbb{Z})^n$, so the representation cannot be faithful if $m > n$; contradiction. 
