By a theorem of Keune (which is more of definition rather then theorem per se) K-groups of ring $R$ are Puppe's derived functors of pronilpotent completion of $GL(R)$. They are derived in the most straightforvard sense just as in case of abelian category: in absence of projective resolutions we have hunky dory simplicial ones. For any group $X$ viewed as degenerate simplicial group, taking Kan loops $G$ of classifying space $BX$ we functorially obtain its simplicial resolution $G B X$ with counit augmentation to $\pi_0(GBX) = X$ and higher homotopy groups vanishing. With this knowledge, we can upgrade any unital endofunctor $F: Grp \to Grp$ to its derived version $\mathcal L_{\bullet}F := \pi_{\bullet}(F(GBX))$. Keune's theorem follws directly from the facts that 1) +-construction is a homological equivalence and 2) all H-spaces, and $BGL(R)^+$ in particular, are $\mathbb Z$-local in Sullivan sense; apply Bousfield-Kan construction of pro-$p$-completion via $p$-nilpotent completion (in our case $p = 0$) of Kan loops and we're done.

Take a group $G$, and complete its rational group algebra $\mathbb QG$ in augmentation ideal. Define Malcev completion $\widehat G$ as group of grouplike elements in this complete Hopf algebra. It is, indeed, a functor and it is unital. Let's take a closer look at $\mathcal L_{\bullet} (\widehat{GL(R)})$.

So, what is this "rational homotopy K-theory"? It's probably wide known and (probably) related to Milnor K-groups.

Related question: can we say something about unstable version of these, i. e. $\mathcal L_{\bullet} (\widehat{GL(n, R)})$? It looks more computable, and for $GL(2, \mathbb Z)$ we have particularly nice resolvent of Gruenberg type obtained from combinatorial presentation. (Sidenote musings: shouldn't it resemble rationalisation of ordinary K-theory — at least in low degrees — for infinite fields?)