It is probably a bad idea to try to compute the Cartier dual but better to let
Cartier do that for you... If $G$ is a flat commutative finite group scheme with
affine algebra, the commutative and cocommutative $A$ which is the flat over the
base ring $R$. Then the Cartier dual is the spectrum of the dual Hopf algebra
$A^\ast$ of $A$. The proof of this is simple enough; an $R$-algebra homomorphism
$A\rightarrow R$ corresponds to a $\varphi\in A^\ast$ of multiplicative type,
$\Delta^\ast(\varphi)=\varphi\otimes\varphi$, which in turn corresponds to a Hopf
algebra map $R[t,t^{-1}]\rightarrow A^\ast$. As this can be done for all
$R$-algebras we get an isomorphism of functors.

Doing this for $\alpha_p$ which has $A=R[x]/(x^p)$ we get that $A^\ast$ has a
basis dual to $x^i$ of the form $1/i!\partial^i/\partial x^i$. Unravelling the
definitions one gets the formula $s\mapsto(t\mapsto \exp_{p-1}(st))$.