Regarding the question:
In particular, given a Hopf algebra $H$, under what conditions can one recover the structure of $H$ from it's group of group-like elements?
the simplest case in which this happens corresponds to f.d. cocommutative hopf algebras. In particular, it can be shown that:
If $H$ is a finite dimensional, cocommutative hopf algebra, over an algebraically closed field of characteristic zero, then:
where $G(H)$ is the set of the grouplikes of $H$, which is a group, and $kG(H)$ is its group algebra.
This is actually a consequence of the Cartier-Konstant-Milnor-Moore theorem (when applied to the finite dimensional case).
Another interesting situation arises in the case of the commutative hopf algebras:
If $H$ is a finite dimensional, commutative hopf algebra, over an algebraically closed field of characteristic zero, then:
where $H^*$ is the dual hopf algebra of $H$, $G(H^*)$ is the set of the grouplikes of $H^*$, $kG(H^*)$ its group hopf algebra and $\big(kG(H^*)\big)^*$ the dual hopf algebra of the group hopf algebra $kG(H^*)$.
A special case combining both commutativity and cocommutativity is discussed in Reference on correspondence between commutative Hopf Algebras and Groups. Maybe that could also be of some interest for the OP.