The <PERSON>-Sham approach to density functional theory is based on a self-consistency scheme to obtain the ground-state electron density of the investigated material. As a starting point the scheme uses an input density that is some approximation to the ground-state density. It uses the density to determine the effective potential $V_\text{eff}(\mathbf{r})$ consisting of the Hartree potential, the exchange-correlation potential, and the external potential due to the atomic nuclei.
When you have this the next step is the solving of the <PERSON>-Sham equations (here in Hartree atomic units) $$\left. -\frac{1}{2}\nabla^2+V_\text{eff}(\mathbf{r}) \middle| \psi_i \right\rangle = \left. \epsilon_i \middle| \psi_i \right\rangle,$$ where the $\psi_i(\mathbf{r})$ are the Kohn Sham orbitals. The Kohn-Sham equations are eigenvalue equations that are typically approached by introducing some kind of basis functions $\phi_i(\mathbf{r})$ to get a formulation of the problem that can be solved with the tools of linear algebra. With the basis functions we obtain a Hamiltonian matrix $$H_{ij} = \left\langle \phi_i \middle| -\frac{1}{2}\nabla^2+V_\text{eff}(\mathbf{r}) \middle| \phi_j \right\rangle$$ and an overlap matrix $$S_{ij} = \left\langle \phi_i \middle| \phi_j \right\rangle.$$ The Kohn-Sham equations thus become a generalized eigenvalue matrix problem. It can be solved by turning it into a standard eigenvalue problem and then diagonalizing the matrix for this problem with standard linear algebra methods. 768
In the end you obtain the orbital energies along with the eigenstates of the problem in terms of linear combinations of the basis functions.
Next would be the determination of an output density with these eigenstates by occupying them with electrons. Then you have to guess an improved input density based on the last input densities and the output density. The loop is iterated again and again until the density is self-consistent, i.e., input and output densities are (nearly) the identical.
You mentioned in your question that you already have the electron density, but this is probably only an approximation to the self-consistent density that you obtain from the Kohn-Sham scheme. Otherwise you should have constructed the Kohn-Sham orbitals in the determination of your density.
Note that the choice of the basis set $\left\lbrace\phi_i\right\rbrace$ is critical to obtain reasonable results. If you want to include all electrons in your calculations the orbitals feature strong oscillatory behaviour near the atomic nuclei because of the singularity of the external potential at these points. You need a lot of intuition to design a basis that is capable to represent such orbitals. An example for a basis that is capable to do this is the linearized-augmented plane-wave (LAPW) basis.
Another approach would be the introduction of the pseudopotential approximation that cuts away the core electrons and replaces the real external potential by a pseudopotential that yields smooth valence electron eigenstates featuring (for a prototype system) the same energy eigenvalues as an all-electron calculation. These can be represented by a simple plane-wave basis but the transferability of the pseudopotential to materials deviating from the prototype system is not necessarily given. 976