Partial answer: 

For the case of cocommutative Hopf algebras, over an algebraically closed field $k$, with $char(k)\geq 3$ you can have a look at:  

[Modules of solvable infinitesimal groups and the structure of representation-finite cocommutative Hopf algebras][1], R.Farnsteiner, D.Voigt, Math. Proc. Cambridge Philos. Soc., v.127, p.441-459, 1999  

and the references therein (among them, there is also an interesting older paper by the same authors discussing the case of cocommutative hopf algebras of finite representation type, over an algebraically closed field of $char(k)>0$). 


  [1]: https://www.semanticscholar.org/paper/Modules-of-Solvable-Innnitesimal-Groups-and-the-Farnsteiner-Voigt/a7ffd5538cd3e73320773c05ba4311e12e12905d