Not really an answer, but too long to be a comment.

I have thought about this question before, without much progress. It seems difficult. Even just to recognize if a manifold has the structure of a Lie group seems to be a difficult problem. And a manifold can be a Lie group in different ways.

For instance, the exponential map from the Lie algebra $\mathfrak{n}$ of strictly upper triangular $n \times n$ matrices is a diffeomorphism with the group $N$ of unipotent upper triangular matrices, so $N$ has its natural Lie group structure as well as the abelian Lie group structure obtained by transporting the addition from $\mathfrak{n}$ through the exponential map.

Thinking about it algebraically, asking for a manifold to be a Lie group is the same as asking for its function algebra to be a Hopf algebra (with some topology, perhaps), and the responses to this question seem to indicate that it is not very easy to recognize a Hopf algebra from the algebra structure alone.

If you want a homogeneous space rather than a Lie group, you are then asking for the function algebra to be a (right or left) coideal subalgebra of a Hopf algebra. That seems even more difficult than looking for a Hopf algebra structure. And again, there can be no canonical construction because the same manifold can be a homogeneous space of different Lie groups: for example, $S^2$ is a homogeneous space of $SU(2)$, $SL(2,\mathbb{C})$, $O(3)$, and $SO(3)$.

So I am pessimistic about there being a nice characterization, or even about the existence of simply stated sufficient conditions. But I'll be watching this thread and hoping that somebody comes up with something!

Edit: Oh yeah! I forgot to mention something. Thinking further about the problem of identifying a Lie group, given only the manifold structure: the multiplication map on a Lie group $G$ turns the cohomology ring of $G$ into a graded Hopf algebra. I don't know the details, but my understanding is that (perhaps with some assumption on the cohomology groups being finitely generated free modules over the coefficient ring?) that if the cohomology ring of a manifold has the structure of a graded Hopf algebra, then the manifold is an $H$-space, i.e. it has a multiplication and a unit map, but it is not necessarily associative. For instance $S^7$ is an $H$-space, but not a Lie group. So cohomology gets you a certain distance, but not all the way.