Maybe this can be of use: James S. Milne, Algebraic Groups, Lie Groups, and their Arithmetic Subgroups. 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.
darij grinberg
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