Hartree-Fock vs. DFT vs. Hohenberg-Kohn vs. Kohn-Sham What is the difference between these four approaches? 

My current understanding is that Hartree-Fock uses only 1 slater-determine (accounting for antisymmetry) but neglects correlation between electrons. HF orbitals are primarily determined self-consistently. 

DFT is the idea that there is a functional (presently unknown) that maps energy in terms of density. Hohenberg and Kohn found a proof that it exists but did not actually find it. 

$E[p] = T[p] + E(EE)[P] + E(EN)[P]$

I am rather confused about where kohn-sham theory fits into all of this. Is it's entire purpose simply to define kinetic energy? How does it do this?
The Kohn-Sham approach is one on top of the original DFT idea proposed by Hohenberg-Kohn.
They propose – to find the energy functional and its equations, which, as you said, are unknown – to start with what is known: the functional if there is no interactions. So that is what they do: approximate the kinetic energy and Slater determinant from the known, non-interacting system and add all the terms that are additionally known (i.e. Coulomb, kinetic energies and the electron/nucleus potential) until you are left with the Kohn-Sham equation, which has &quot;the smallest unknown&quot;, the exchange &amp; correlation potential.
I recommend to read more in a book or on Wikipedia in case you want a more detailed picture on any/all of the methods.