lie algebra cohomology - Chevalley Eilenberg complex definitions?

The first complex, from Weibel, is a projective resolution of the trivial $mathfrak g$-module $k$ as a $mathcal U(mathfrak g)$-module; I am sure Weibel says so!

Your second complex is obtained from the first by applying the functor $hom_{mathcal U(mathfrak g)}(mathord-,k)$, where $k$ is the trivial $mathfrak g$-module. It therefore computes $mathrm{Ext}_{mathcal U(mathfrak g)}(k,k)$, also known as $H^bullet(mathfrak g,k)$, the Lie algebra cohomology of $mathfrak g$ with trivial coefficients.

The connection with deformation theory is explained at length in Gerstenhaber, Murray; Schack, Samuel D. Algebraic cohomology and deformation theory. Deformation theory of algebras and structures and applications (Il Ciocco, 1986), 11--264, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 247, Kluwer Acad. Publ., Dordrecht, 1988.

In particular neither of your two complexes 'computes' deformations: you need to take the projective resolution $mathcal U(mathfrak g)otimes Lambda^bullet mathfrak g$, apply the functor $hom_{mathcal U(mathfrak g)}(mathord-,mathfrak g)$, where $mathfrak g$ is the adjoint $mathfrak g$-module, and compute cohomology to get $H^bullet(mathfrak g,mathfrak g)$, the Lie algebra cohomology with coefficients in the adjoint representation. Then $H^2(mathfrak g,mathfrak g)$ classifies infinitesimal deformations, $H^3(mathfrak g,mathfrak g)$ is the target for obstructions to extending partial deformations, and so on, exactly along the usual yoga of formal deformation theory à la Gerstenhaber.

By the way, the original paper [Chevalley, Claude; Eilenberg, Samuel
Cohomology theory of Lie groups and Lie algebras.
Trans. Amer. Math. Soc. 63, (1948). 85--124.] serves as an incredibly readable introduction
to Lie algebra cohomology.