It is difficult to give a precise definition, because there are many cohomology theories which go by the name of BRST. It might be helpful to give a couple of examples, not necessarily in chronological order.
In classical (i.e., non-quantum) physics, there is a notion of BRST cohomology which encodes symplectic reduction à la Marsden-Weinstein. This was the subject of my answer to an earlier question. It is the cohomology of a double complex and one of the differentials is the Chevalley-Eilenberg differential. However, it does admit generalisations (e.g., to coisotropic reduction) where there is no Chevalley-Eilenberg differential, while still being referred to as BRST cohomology.
The original BRST cohomology arose in quantum field theory, where it arose as an "invariance" of the gauge-fixed Fadde'ev-Popov action for a gauge theory and plays an important role in proving the renormalisability of four-dimensional gauge theories. The BRST differential again has a part which is the Chevalley-Eilenberg differential of the Lie algebra of the gauge group. Again, there are generalisations (to theories with "open algebras") where there is still a BRST cohomology, but now there is no longer any Chevalley-Eilenberg differential.
In the context of two-dimensional conformal field theories (e.g., string theory), the BRST cohomology can be identified with a certain (relative) semi-infinite cohomology in the sense of Feigin.
In all cases precise definitions can be given, but they are different.