Consider the universal enveloping algebra $U(\mathfrak g)$ of your Lie algebra $\mathfrak g$. Since it is a Hopf algebra, then you can construct a filtered simplicial cocommutative coalgebra $A_\bullet$:
$A_i=U(\mathfrak g)^{\otimes i}$
face maps are given by applying the product
degeneracy maps are given by applying the unit
The $E_1$ term of the associated spectral sequence is precisely the Chevalley-Eilenberg chain complex*.
In other words, the Chevalley-Eilenberg complex of $\mathfrak g$ is a by-product of the Bar complex of $U(\mathfrak g)$. And the Bar complex of an augmented unital algebra $A$ arises as the chain complex associated to the simplicial set $Nerve(A)$ (where I view $A$ as a linear category with one object).
- this is another way of saying what Mariano Suárez-Alvarez says in his answer.