[Skip down to the bottom for a correction] Let's work over a field k, assume it is as nice as you need it to be.. Suppose I have an ordinary (edit: commutative) affine group scheme G = Spec(A) over k, I may produce from G the following objects:
1.) A simplicial ring k(A,n) having homotopy groups equal to A in degree n and k elsewhere with all (co)degeneracy maps being k-linear. Probably the most obvious choice for a notion of an Eilenberg-MacLane space. Let Spec(k(A,n)) be the associated derived scheme.
2.) We can also associate to X the (iterated...) classifying stack K(G,n). I'm not fully sure how such a thing is constructed for n>1. (is this even a remotely nice object? or are we talking (n,infinity)-cats here).
To what extent are these objects known to be 'the same'. To be concrete, let n = 1 and let's focus on the following case
1'.) k(A,1) is a simplicial ring with non-trivial homotopy concentrated in degree 1. You can very directly construct it by starting with the constant simplicial ring associated to k, then adding and killing off generators in higher degree.
2'.) K(G,1)(X) is of course, the classifying stack of principle G-bundles on a scheme X. In any case, that's what I wish K(G,1) is... (though, morally, I should probably be looking at the classifying space of principle G-bundles on the derived scheme Spec(A*) where A* is the constant simplicial ring associated to A.
Second edit: Qiaochu Yuan's comment below shows that the first construction above does not make sense as stated. I guess what I really want to know is this: Is there a derived scheme playing the role of the n-th Eilenberg MacLane space to the group scheme associated to the multiplicative group of a field and does it homtopically classify the objects one hopes it does (n = 0 characters on the scheme, n = 1 line bundles, n =2 azumaya algebras maybe??).