As Henry Wilton asked, it would be nice to know what something you would like to know.
If you restrict to $GL_+(k,\mathbb R)$, meaning elements with positive determinant, then
you can split the matrices as positive multiple of the identity times a matrix of determinant
1, so a good place to start is to look at homomorphisms to $SL(k,\mathbb R)$.

In general, the space of homomorphisms up to conjugacy is not Hausdorff. For example,
when $k=2, n=1$, the matrices
$((1,a),(0,1))$ are all conjugate for $a \ne 0$, but they limit to the identity matrix.
But most of these quotients are Hausdorff, and in any case one can take the Hausdorffication.
For $k=2$, the trace of an element of $SL(2,\mathbb R)$ is a complete invariant of the
Hausdorfication; in general, you need the coefficients of the characteristic polynomial,
or equvialently, the sequence of traces of the first $n-1$ powers to parametrize the Hausddorffication of the conjugacy
class of an element of $SL(n,\mathbb R)$.

For $k=2, n=2$, if the generators of the free group map to elements $A$ and $B$,
then the traces of $A, B, AB$ are enough to determine the Hausdorffication of the set
of these homomorphisms. The trace of any other element in the group they generate is
a polynomial in these three traces, which can be chosen arbitrarily and independently.
These can be found using the trace relation
$$T(AB) + T(A B^{-1}) = T(A) T(B) $$
(This can be derived from the fact
that $A$ satisfies its characteristic equation, $A^2- T(A) A + I = 0$, multiplying on the right by
$A^{-1} B$ and taking the trace.)

For $k=2, n=3$, the traces of $A, B, C, AB, BC, CA, ABC$ are enough to determine the
conjugacy class, but the trace of $ABC$ satisfies a quadratic equation with coefficients
that are polynomials in the other traces. (The second root is the trace of $ACB$).

For general $k$ and $n$, the ring generated by all traces of all elements of the group
(as a function on the space of homomorphisms from the free group to $SL(n,\mathbb R)$,
or to $GL(n, \mathbb R)$
is generated by the traces of a finite subset of the elements; it's something people have studied, it gets more and more complicated, but I don't know the details.