Hi, its seems that your question is rather soft, so I will give some arguments, which come close to a description you seem to search for.

I'll start with Clifford theory, Kirillov orbit method or the Mackey Machine in the case of compact groups. I will not use the notation $K_0$, since you can argue with irreducible representations directly in this context, so with the generators of $K_0$. I learnt this theory from this paper: http://arxiv.org/abs/0807.4684. It's theorem 2.1.

Let $G$ be a compact group and $N$ a closed, normal subgroup, then $G$ acts by conjugaction on $N$, hence on the "set" of its irreducible representations. Let $\pi$ be an irreducible representation of $G$.

Facts:

- Cliffords theorem: $Res_N \pi$ contains exactly one $G$-orbit $\{ \sigma \}$ of irreducible representations of $N$.
- Let $G_\sigma$ be the stabilizer of $\sigma$, then we have a one-to-one correspondance
of irred. reps. $\pi$ of $G$, which contain $\sigma$ in $Res_N \pi$, and irred. reps $\pi' $ of $G_\sigma$, which contain $\sigma$ in $Res_N \pi$, by the induction
$$ \pi' \mapsto Ind_{G^\sigma}^{G} \pi.$$

This theory is in particular helpful, when $N$ is abelian. In fact, it will work equally well for finite index or cocompact normal groups, which are type 1. In general you should expect a direct integral.

To sum up, we have an isomorphism

$$ \bigoplus_{\{ \sigma \} } K_0( G^\sigma) \cong K_0(G)$$

Further comments mostly for reductive groups over local fields and Lie groups (don't take them to serious though):

The Mackey machine is available in the case of unitary representation of locally compact separabel groups, where $N$ is a type I group. It is Mackey's paper on group extensions. I think that it is hard to single out the finite dimensional representations.

The finite dimensional representation of a reductive Lie group are glued together from finite dimensional representations of the maximal compact subgroup, but most of them are not unitary (Weyl's unitary trick). For $SL_2(\mathbb{R})$, the only unitary, finite dimensional representation is trivial. For the general linear group they factor through the determinant. So I guess, you might want to drop unitarizability here. All finite dimensional representation of reductive Lie groups appear as sub-module, or sub-quotient of parabolic induced representations, and are almost never unitary.

I recall that there is a statement that the unitary representation theory of a general Lie group depends essentially on the representation theory of reductive group and nilpotent groups. I know that this is a rigorous statement for algebraic group, since there every group is an extension of a reductive one by a unipotent one.

I'll give one concrete example here: Consider the upper triangular matrices of $\mathbb{R}$. All finite dimensional unitary representations will be unitary representations of the diagonal matrices. In general, it seems that you only have to answer the question for reductive groups.

Also index theory has been used to construct discrete series by Atiyah-Schmid, but I do not know much about that. For $SL_2(\mathbb{R})$, knowing in which parabolic induction to find the finite-dimensional reps is equivalent to finding the discrete series reps. Discrete series are the "prime objects".