Note that Coulomb’s Law is experimentally determined, though because Maxwell’s equations describe all electromagnetic phenomena you can consider doing it this way (so you can see how the constant $k$ comes about):
Consider this equation which is one of Maxwell’s equations,
$$\oint_S \vec E \cdot \vec{dS} = \frac{Q}{\epsilon_0}$$
where this surface integral $S$ is taken over a closed sphere surrounding the charge $Q$ then
$$E (4\pi r^2) = \frac{Q}{\epsilon_0}$$
and given that the electric field $E$ is defined as the force per unit charge ie.,
$$E = \frac{F}{q}$$
then we can write
$$F = \frac{Qq}{4\pi \epsilon_0 r^2}$$
which is Coulomb’s Law.
The vacuum permitivity constant is a measure of how much a vacuum permits an electric field. Most problems we assume that there is nothing else in the space between the charges we are studying, which is why we use $\epsilon_0$. In cases where we do, we use the dielectric constant $\epsilon$ which is a measure of how much the medium permits an electric field.
When we manipulate Maxwell’s equations, we end up with the electromagnetic wave equation and as you can see in the link
$$v = \frac{1}{\sqrt{\mu \epsilon }}$$
where $v$ is the speed of an electromagnetic wave in a region with permeability $\mu$ and permitivity $\epsilon$. Of course if we are in a vacuum this becomes
$$c = \frac{1}{\sqrt{\mu_0 \epsilon_0}}$$
where $c$ is the speed of light in a vacuum.
