Since I am not an expert in algebraic groups; Is there a description of algebraic groups with semisimple category of finite dimensional representations such that they have only finitely many irreducible representations? Maybe there is a classification?

These are the finite linearly reductive groups. In characteristic zero, they are just the finite algebraic groups. In positive characteristic, Nagata's theorem gives the following characterization of linearly reductive groups over a field of characteristic $p$: $G$ is linearly reductive if and only if there is a finite index subgroup $H$ that is a subgroup of a torus, and the quotient $G/H$ has order prime to $p$. For our purposes, it is therefore necessary and sufficient that the diagonalizable normal subgroup $H$ be finite, and the quotient $G/H$ be finite étale and tame. By Lemma 2.11 of Tame stacks in positive characteristic (this paper also generalizes to arbitrary base schemes), there is a finite purely inseparable extension $k'$ of the base field such that $G_{k'}$ is isomorphic to the semidirect product $H_{k'} \rtimes (G/H)_{k'}$. You can also find a discussion in DemazureGabriel Chapter IV section 3, Theorem 3.6. 


This type of question has probably come up previously on MO, but in any case I'd suggest one especially natural example arising in connection with semisimple algebraic groups or semisimple Lie algebras. This involves socalled "fusion rules", and one of several approaches gets a nice overview in a paper by H.H. Andersen and his student J. Paradowski Fusion categories arising from semisimple Lie algebras, Roughly speaking, the idea is to start with ordinary tensor products of finite dimensional highest weight modules, but then truncate in a certain way that leaves a semisimple category with only finitely many simple objects. Following Ringel, Donkin, Mathieu, one gets a theory of "tilting modules" for semisimple algebraic groups in prime characteristic. 

