Why does the amount of dark matter increase the further away from the galaxy's center? Why isn't the dark matter just randomly distributed? How does it know where to go?
 A: If you take a dust cloud and let it collapse under it's own gravity then it will not form a compact mass unless there is some way for it to lose energy. Unless the dust particles can lose energy they just fall into the central potential then zoom past the centre and back out again. So the dust cloud just oscillates about the centre of mass.
The principal source of energy loss is through electromagnetic interactions. Dust particles collide and transfer kinetic energy to lattice vibrations, i.e. their temperature rises, and they emit photons by black body radiation. The emitted photons carry away energy.
However dark matter particles do not interact via the electromagnetic force. We don't know exactly what they are, but it's widely believed that they interact only via the weak force. This means they are much, much less efficient at dissipating energy than normal baryonic matter, and consequently a cloud of dark matter will collapse much more slowly than a cloud of baryonic matter.
We expect the density profile of dark matter to be roughly comparable to the density profile or baryonic matter, so it is densest at the centre and falls off as some power law with distance from the centre of the cloud. However the rate of decrease of density will be much slower than that of baryonic matter and therefore we expect a galaxy to be embedded in a much large cloud of dark matter.
So the dark in not randomly distributed, as you suggest in your question, it just extends out to distances well past the edge of the galaxy.
There are lots of papers around modelling dark matter structure around galaxies. A quick Google found this paper, which gives quite a nice description. The cumulative mass distribution, i.e. the mass within a radius $r$, varies with $r$ as shown in this diagram from the paper:

This shows measured and calculated mass distributions for the Andromeda galaxy. The edge of the visible matter is around 40 kpc, and the dashed blue line shows how the mass of visible matter stops increasing outside this radius. However the total mass carries on increasing, which must be due to dark matter lying outside the galaxy. The black dots are measurements deduced from the rotation curve.
A: There seems to be some confusion here. The density distribution of dark matter increases towards the centre of a galaxy. One of the usual approximations is that it follows a Navarro, Frenk & White profile.
$$ \rho(r) = \frac{\rho_0 (R_s/r)}{\left( 1 + r/R_s\right)^2},$$
where $\rho(r)$ is the density at radius $r$, and $R_s$ is a scale length for the model. You can see that the density becomes very large when $r$ is small. This is an empirical approximation to N-body simulations of how dark matter behaves under the assumption that dark matter only interacts gravitationally. i.e. It is just gravity that tells dark matter how to behave.
The mass between any range of radii is then found by integrating the profile over volume
$$M = \int^{r_2}_{r_1} \rho(r)\ 4\pi r^2\ dr$$
i.e. the mass contained within a shell of thickness $dr$ is given by
$$dM= \frac{4\pi R_s r}{\left( 1 + r/R_s\right)^2}\ dr$$
i.e this behaves like $f(x) = x/(1+x)^2$ and the mass in a shell of a given thickness increases until $r=R_s$ and then declines at larger radii.
So, if dark matter halos really are represented by a NFW profile, then the statement implicit in your question is not true. However what is true is that normal matter appears to be more concentrated towards the centre of galaxies. This is because, in the cold-dark-matter (CDM) model, it "falls" into the pre-existing gravitational potentials that were created by dark matter early in the universe. (CDM can clump earlier than normal matter because it is unaffected by gas and radiation pressure). As the normal matter falls into the potential created by the dark matter it "virialises". That is, collisions between particles dissipate energy (i.e. the gas cools), such that statistically the average kinetic energy of a particle is half its gravitational potential energy. This cause it to "sink" into the dark matter halo. The canonical work to read is White & Rees (1978)
