  The purpose of this paper is to define an $\alpha$-type cohomology, which we
call $\alpha$-type Chevalley-Eilenberg cohomology, for Hom-Lie algebras. We
relate it to the known Chevalley-Eilenberg cohomology and provide explicit
computations for some examples. Moreover, using this cohomology we study formal
deformations of Hom-Lie algebras, where the bracket as well as the structure
map $\alpha$ are deformed. Furthermore, we provide a generalization of the
Grand Crochet and study, in a particular case, the $\alpha$-type cohomology for
Hom-Lie bialgebras.
