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.