We formulate a Hartree-Fock-LAPW method for electronic band structure calculations. The method is based on the Hartree-Fock-Roothaan approach for solids with extended electron states and closed core shells where the basis functions of itinerant electrons are linear augmented plane waves. All interactions within the restricted Hartree-Fock approach are analyzed and in principle can be taken into account. In particular, we have obtained the matrix elements for the exchange interactions of extended states and the crystal electric field effects. In order to calculate the matrix elements of exchange for extended states we first introduce an auxiliary potential and then integrate it with an effective charge density corresponding to the electron exchange transition under consideration. The problem of finding the auxiliary potential is solved by using the strategy of the full potential LAPW approach, which is based on the general solution of periodic Poisson's equation. Here we use an original technique for the general solution of periodic Poisson's equation and multipole expansions of electron densities. We apply the technique to obtain periodic potentials of the face centered cubic lattice and discuss its accuracy and convergence in comparison with other methods.