For me the most natural examples to think about are those coming from linear algebraic groups in prime characteristic, in the form of *Frobenius kernels*.   These are extensively treated in Jantzen's book *Representations of Algebraic Groups*, with foundations laid in Part I.    Concretely, you can start with the restricted universal enveloping algebra of the Lie algebra of such a group and associate to it naturally a finite group scheme.    This is only the "first" Frobenius kernel.

Examples of this kind are "affine" but not linear algebraic groups.