  The explicit semiclassical treatment of logarithmic perturbation theory for
the bound-state problem within the framework of the radial Klein-Gordon
equation with attractive real-analytic screened Coulomb potentials, contained
time-component of a Lorentz four-vector and a Lorentz-scalar term, is
developed. Based upon $\hbar$-expansions and suitable quantization conditions a
new procedure for deriving perturbation expansions is offered. Avoiding
disadvantages of the standard approach, new handy recursion formulae with the
same simple form both for ground and excited states have been obtained. As an
example, the perturbation expansions for the energy eigenvalues for the
Hulth\'en potential containing the vector part as well as the scalar component
are considered.
