The $K_2$ functor has an important role in representation theory, starting from
works of Bloch in which he identified the Kac-Moody central extension of loop algebras in terms of $K_2$ - a more updated form of this is the IHES paper of Deligne-Brylinski about $K_2$ central extensions of reductive groups. The universal class in $H^4(BG,Z)$ for a simple group $G$ which is responsible for the level in Kac-Moody algebras/Chern-Simons theory or the $q$ in quantum groups etc can be interpreted as representing this $K_2$ central extension. Anyway it's a very beautiful story we haven't really fully seen the impact of yet (but is also behind eg the famous constructions of Fock-Goncharov in Teichmuller theory).

Anyway that's not the kind of role you're talking about, in which we take higher K groups of a category of representations. One way to answer that is to say certainly the full homotopy theory of a category of representations is very important - but there we usually ask for something more subtle, ie identifying the entire category itself in some geometric way rather than just its $K$-theory spectrum or the homotopy groups of the latter. To really see higher K-groups arising the way you're asking I think you'd need to consider problems in which FAMILIES of representations play a central role -- single representations (or contractible families thereof) are measured by $K_0$, but if you are interested in measuring families over a circle (ie automorphisms) or over more complicated bases (eg higher spheres - i.e. $n$-automorphisms), these would be measured by invariants in $K_1$ or higher $K$-groups.

[Edit: When I said "something more subtle" I meant the following: n equivalence of (enhanced) derived categories gives rise to an isomorphism of K-theories -- eg Beilinson-Bernstein localization identifies K-groups of categories of Lie algebra representations with those of flag varieties. This certainly doesn't mean in general we know explicitly the K-groups! eg there are few categories we understand as well as $Z$-modules, but the corresponding K-groups are complicated! However it seems that these subtleties are arithmetic, and I don't see a clear representation theoretic role for them, though would be happy to learn otherwise.]

The two other things that come to mind from your question are

in Jacob Lurie's survey article on elliptic cohomology (available here) there's a theorem describing the K-theory spectrum of categories of integrable loop group representations in terms
of a DAG version of nonabelian theta functions (in the spirit of results of Kac-Peterson, Looijenga and Ando)

Beilinson's work on epsilon factors explains algebraic K-theory beautifully and applies it in a context related to automorphic forms.