Maybe this can be of use: [James S. Milne, *Algebraic Groups, Lie Groups, and their Arithmetic Subgroups*][1]. Chapter I of this text considers affine algebraic groups; these are in one-to-one contravariant correspondence with finitely generated commutative Hopf algebras. Hopf algebras that happen to be group algebras correspond to diagonalizable algebraic groups; these are considered in I.12. Unfortunately, I don't see any results on how to recognize a diagonalizable algebraic group, but maybe this viewpoint can help you in googling.

  [1]: http://jmilne.org/math/CourseNotes/ala.html