What makes skew characters of the symmetric group special? For integer partitions $\mu\subset\lambda$ we can define the skew character $\chi^{\lambda/\mu}$ (for example?) via the Littlewood-Richardson rule.
Many combinatorial gadgets and algorithms extend in a very natural way from the case of partitions to the skew case.
I think it is fair to expect that characters corresponding to partitions are special, because they are irreducible.
I would like to know whether there are (representation theoretic) properties of skew characters that make them special.
 A: Here are a few references to expand on my comment about "convexity" properties of skew Schur functions $s_{\lambda/\mu}$.
As the OP points out, it's maybe most natural for the question to assume that someone has handed you the decomposition of $s_{\lambda/\mu}$ into irreducibles, i.e., they've given you a list of LR coefficients. For a "convexity" conjecture on LR coefficients (which could be translated to say something about the list you've been given) see Conjecture 1 (due to Lam-Postnikov-Pylyavskyy) of https://arxiv.org/abs/math/0608134. Many special cases of that conjecture are known- see the references given in that paper.
But another thing you can do is think of $s_{\lambda/\mu}$ as a polynomial (i.e., do the monomial expansion), and ask questions about its convexity properties. For instance, it is known that the nonzero coefficients of this polynomial are exactly the lattice points of some convex polytope (this is called the "saturated Newton polytope" (SNP) condition); see https://arxiv.org/abs/1703.02583. It is also conjectured that the "normalized" version of $s_{\lambda/\mu}$ is "Lorentzian" (another kind of convexity property) in https://arxiv.org/abs/1906.09633.
These kind of convexity properties are likely enough to tell the character $s_{\lambda/\mu}$ apart from a "random" character of a symmetric group; but it is doubtful they would lead to a procedure for exactly recognizing $s_{\lambda/\mu}$. For instance, the Stanley symmetric functions (mentioned in a previous answer), also satisfy the SNP property (and indeed this is how the SNP property is deduced for skew Schur polynomials).
A: One possible answer is that skew characters are not really the largest extension,
but a special-case of certain characters associated with any subset of boxes in the plane. For example, if this subset is the Rothe diagram of a permutation,
the character is a Stanley symmetric function.
It is an open problem, see 
[Ricky Ini Liu, Specht modules and Schubert varieties for general diagrams. Massachusetts Institute of Technology. 2010] ,
to describe the Schur expansion of these (generalized Specht modules) combinatorially.
A quick (quick to implement) way to compute the Schur expansion of Stanley symmetric functions, is to sum over all reduced words of the given permutation, and then take the descent set of the reverse word.
The Schur expansion is then
$$
F_w(x) = \sum_{a \in RW(w)} s_{comp(DES(rev(a))}(x)
$$
where $rev(a)$ is the reverse of the reduced word,
$DES$ is the descent set, $comp$ is the classical map from subsets of $[n-1]$ to compositions, and a Schur polynomial indexed by a composition is computed via the first Jacob-Trudi identity.
The following code also works in Sage:
G = SymmetricGroup(6)
w = G.from_reduced_word(Permutation([2,4,6,1,5,3]).reduced_word())
f = w.stanley_symmetric_function()
s = SymmetricFunctions(QQ).schur()
s(f)

This example outputs:
s[3, 2, 1, 1] + s[3, 2, 2]

