In Weibel's An Introduction to Homological Algebra, the Chevalley-Eilenberg complex of a Lie algebra $g$ is defined as $\Lambda^*(g) \otimes Ug$ where $Ug$ is the universal enveloping algebra of $g$. The differential here has degree -1.
I have been told that the Chevalley-Eilenberg complex for $g$ is
$C^*(g) = \text{Sym}(g^*[-1])$,
the free graded commutative algebra on the vector space dual of $g$ placed in degree 1. The bracket $[,]$ is a map $\Lambda^2 g \to g$ so its dual $d : = [,]* \colon g^* \to \Lambda^2 g^*$ is a map from $C^1(g) \to C^2(g)$. Since $C^*(g)$ is free, this defines a derivation, also called $d$, from $C^*(g)$ to itself. This derivation satisfies $d^2 = 0$ precisely because $[,]$ satisfies the Jacobi identity.
Finally, Kontsevich and Soibelman in Deformation Theory I leave it as an exercise to construct the Chevalley-Eilenberg complex in analogy to the way that the Hochschild complex is constructed for an associative algebra by considering formal deformations.
The first is a chain complex, the second a cochain complex, and what do either have to do with formal deformations of $g$?
Best Answer
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.