2 added a remark on Lie - Kolchin

Even when viewed as an additive category, Rep(G) is not semisimple, so it's not really clear to me what such a description would entail... But simple objects in it are (in a different language) irreducible representations of a semidirect product of groups and Mackey theory was invented precisely with the goal of determining them. The answer, in short, is that each simple G-module has the form $\operatorname{Ind}_{L}^{G}(\sigma^{\prime})$, where $\sigma$ is a simple $H$-module, $K_\sigma$ is the stabilizer of $\sigma$ in $K$ (i.e. consists of all elements $k\in K$ s.t. the action of $k$ on $H$ conjugates $\sigma$ into an isomorphic module), $L=K_{\sigma}H<G$ and $\sigma^{\prime}$ is the natural extension of $\sigma$ to $L$. The catch is that the usual Mackey theory applies to unitary representations (which can be infinite-dimensional) of topological groups. Nevertheless, induction functor is defined in the algebraic setting and I think that the "Mackey machine" works for formal reasons (this must be described in Jantzen, but I don't have it close at hand to check).

Returning to the full category Rep(G), there are various filtrations of finite-length representations with simple quotients, and one can take tensor products of filtered objects. However, in general one doesn't expect a manageable description of Rep(G) even in the special situation of a semidirect product of a unipotent group and a torus: this already includes the case when G is the Borel subgroup of a semisimple algebraic group, which has been studied but is not completely understood. Good luck!

Addendum Of course, if G is a solvable connected algebraic group, by Lie Kolchin every irreducible representation is one-dimensional, so simple G-modules are the same as simple G/[G,G]-modules, which have an easy description.

1

Even when viewed as an additive category, Rep(G) is not semisimple, so it's not really clear to me what such a description would entail... But simple objects in it are (in a different language) irreducible representations of a semidirect product of groups and Mackey theory was invented precisely with the goal of determining them. The answer, in short, is that each simple G-module has the form $\operatorname{Ind}_{L}^{G}(\sigma^{\prime})$, where $\sigma$ is a simple $H$-module, $K_\sigma$ is the stabilizer of $\sigma$ in $K$ (i.e. consists of all elements $k\in K$ s.t. the action of $k$ on $H$ conjugates $\sigma$ into an isomorphic module), $L=K_{\sigma}H<G$ and $\sigma^{\prime}$ is the natural extension of $\sigma$ to $L$. The catch is that the usual Mackey theory applies to unitary representations (which can be infinite-dimensional) of topological groups. Nevertheless, induction functor is defined in the algebraic setting and I think that the "Mackey machine" works for formal reasons (this must be described in Jantzen, but I don't have it close at hand to check).

Returning to the full category Rep(G), there are various filtrations of finite-length representations with simple quotients, and one can take tensor products of filtered objects. However, in general one doesn't expect a manageable description of Rep(G) even in the special situation of a semidirect product of a unipotent group and a torus: this already includes the case when G is the Borel subgroup of a semisimple algebraic group, which has been studied but is not completely understood. Good luck!