Akhil,
Let $\mathcal{C}$ be a tensor category, and let $(A,\mu) \in \mathcal{C}-Alg$ be an algebra in $\mathcal{C}$. So $\mu:A\otimes A\to A$ is a morphism in $\mathcal{C}$ and $\otimes$ here means the $\otimes$ in $\mathcal{C}$ (of course there isn't another one around at this point, but I mean to emphasize it's not just the $\otimes$ of Vect).
In this context, it makes sense to talk about $A$-modules in $\mathcal{C}$, whose definition you can guess. These form a k-linear abelian category $D$ with a forgetful functor to $\mathcal{C}$ which forgets the $A$ action.
Now if $\mathcal{C}$ has a fiber functor F, then $\mathcal{C}$ is realized as the Hopf algebra $End(F)$, as you said (well $End(F)^{op}$ I think , but nevermind). The algebra $A$ can be pushed forward by $F$ to an ordinary algebra $F(A)$ in Vect. However, $D$ is not the category of $A$-modules, but a well-known proposition tells that $D$ is the category of $A\rtimes H$-modules, the semi-direct product you asked about.
Notice that no part of the discussion so far asked for any symmetric structure on $\mathcal{C}$, and also $A$ is only an algebra. To define a bialgebra in $\mathcal{C}$, however, one needs $\mathcal{C}$ to be braided, because the compatibility between $\Delta$ and $\mu$ will use the braiding. In your case braiding just means symmetry. I never worked this out in detail, but I imagine that if $A$ is actually a bi-algebra in $\mathcal{C}$, then $D$ gets endowed with a monoidal structure, and that $D$ is the category of $A\rtimes H$-modules, where $A\rtimes H$ is a bi-algebra. Likewise if $A$ is Hopf in $\mathcal{C}$, then $D$ is tensor and $A\rtimes H$ is a Hopf algebra, and $D\cong A \rtimes H$-mod.
Thus ends the part where I'm pretty sure I'm not saying anything too incorrect. Below I will try to answer your actual question. I would not trust it though until somebody smarter agrees with it.
So, now your question becomes (let's revert to considering algebras at the top and not Hopf algebras, since it should be clear how to extend): Given a functor $F:D\to \mathcal{C}$, when is $F$ the forgetful functor corresponding to some algebra $A \in \mathcal{C}$? I think for this, it will be enough to assume (in addition, of course, to assuming that $F$ is faithful and exact) that $D$ has a projective generator $M$ (although maybe this is guaranteed by a lesser assumption?). This is definitely necessary to be able to realize $D$ as some category of modules of a ring, as you desire, and I imagine that you then let $A=\underline{Hom}(M,M)$ (meaning $\mathcal{C}$-internal homs, which are distinct from $Hom_\mathcal{C}(M,M)$!), which will be an algebra in $\mathcal{C}$, and you can plug into the above.
You should definitely read Ostrik's http://arxiv.org/abs/math/0111139 and other papers by Etingof, Nikshych, and Ostrik about fusion and finite tensor categories.
