We consider the significant class of homogeneous CR manifolds represented by some weighted homogeneous polynomials and we derive some plain and useful features which enable us to set up a fast effective algorithm to compute homogeneous components of their Lie algebras of infinitesimal CR automorphisms. This algorithm mainly relies upon a natural gradation of the sought Lie algebras, and it also consists in treating \emph{separately} the related graded components. While classical methods are based on constructing and solving some associated extsc{pde} systems which become time consuming as soon as the number of variables increases, the new method presented here is based on plain techniques of linear algebra. Furthermore, it benefits from a \emph{divide-and-conquer} strategy to break down the computations into some simpler subcomputations. Also, we consider the new and effective concept of comprehensive Gröbner systems which provides us some powerful tools to treat the computations in the much complicated parametric case. The designed algorithm is also implemented in the extsc{Maple} software, what required also implementing a recently designed algorithm of Kapur et al.