General literature does not seem to offer a characterisation of topological groups among all topological spaces. Of course, being completely regular (uniform) is necessary, but separation properties, or indeed any sort of "niceness" like pseudometrisability are not sufficient, since topological groups, for instance, cannot have fixed point property.

There is a homological criterion that is often helpful, to rule out the possibility for a topological space to admit a continuous group structure (even Hspace structure): The rational cohomology ring of a connected topological group (or Hspace) $G$ is a connected gradedcommutative Hopf algebra over $\mathbb{Q}$ and if $H^i(G;\mathbb{Q})$ is finite dimensional for all $i \ge 0$, then, by a theorem of Borel, $H^\ast(G;\mathbb{Q})$ is the tensor product of an exterior algebra on odddimensional generators and a polynomial algebra on evendimensional generators. For example $H^\ast(\mathbb{C}P^n;\mathbb{Q})=\mathbb{Q}[X]/(X^n),\; \deg x=2$ isn't of this form. Hence $\mathbb{C}P^n$ is no topological group. For the theorem (and variations thereof) and further examples see Hatcher: Algebraic Topology, Section 3.C. 


Every topological group is homogenous  this rules out spaces like $[0,1]$, $[0,\omega_1]$ or $\beta \mathbb{N}$. 


@N Unnikrishnan: Let $G$ be a topological group, and $a\ b\in G$. Consider $h:G\rightarrow G$ defined by: $$\forall_{x\in G}\quad h(x) := a\cdot x^{1}\cdot b$$ Then $h$ is a homeomorphism such that $h(a)=b$ and $h(b)=a$. This shows that (in your terminology above) every topological group is strongly homogeneous. In general, the above homeomorphism $h_{a\ b} := h$ is not an involution. Indeed, in general, it is not its own inverse (with respect to composition)actually, $h_{b\ a}$ is the inverse of $h_{a\ b}$ (it is an involution in the Abelian case though since then $h_{a\ b}=h_{b\ a}$). 


This is really a long comment on the answer by Cleft accepted by the OP. Note that there are homogeneous spaces with the fixed point property (e.g. the Hilbert cube) and there are first countable nonmetrizable homogeneous spaces (e.g. Alexandroff's double arrow space). Under additional axioms, there are also homogeneous spaces (even compact ones) which are hereditarily separable but not hereditarily Lindelof and the other way around: some that are hereditarily Lindelof but not hereditarily separable. And the list goes on. So homogeneity is far from characterizing topological groups (even in the class of compact spaces). On the other hand, a (kind of vague) question due to Kunen (I think) is: can one say something interesting about compact right topological groups that cannot be said about compact homogeneous spaces? (besides things like "do not have the fixed point property" or "admits a group operation which is continuous in one variable"). 


A necessary condition for a Hausdorff compact space to admit the structure of a topological group is the Suslin condition (I hope I am using proper terminology)
This is so because Hausdorff compact topological groups admit Haar measure. 

