The form of the Kohn-Sham (KS) exchange potential, which arises from the approximation that the Hartree-Fock (HF) and the exchange-only KS determinant are equal, is derived. Two related procedures to determine the KS exchange potential follow from this approximation: a self-consistent localized HF procedure and a transformation localized HF procedure yielding the local KS exchange potential from HF orbitals. Both procedures can be considered as almost exact exchange KS methods which require only occupied orbitals and are invariant with respect to unitary transformations of the orbitals, i.e., depend only on the first order density matrix. The resulting local KS exchange potentials are free of Coulomb self-interactions and exhibit the correct long-range 1/r-behavior. The Krieger, Li, and Iafrate (KLI) procedure to determine the KS exchange potential can be considered as an approximation to the introduced localized HF procedures. Highly efficient methods to carry out the presented localized HF as well as KLI procedures are introduced. An efficient basis set approach to calculate the Slater potential is presented. The methods can easily be implemented in present standard quantum chemistry codes. Applications to small and medium size molecules and clusters are presented. The Hartree-Fock and the exchange-only KS determinant are found to be surprisingly close. Qualitatively correct, Coulomb self-interaction free KS orbitals and eigenvalue spectra are obtained.