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

I guess that 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 a 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).