I am reconsidering an argument previously I thought obvious but now I feel I do not understand. Let $G$ be a finite flat group scheme over a finite field $k$. $G^\vee$ the Cartier dual of it. Let $TG$ denote its Lie algebra. $\mathbb{G}_a$ the additive group scheme. Then we have
$$ TG\cong \mathrm{Hom}(G^\vee, \mathbb{G}_a) $$
Previously I am thinking $G\cong \mathrm{Hom}(G^\vee,\mathbb{G}_m)$ then take the tangent space we have the argument. But it seems I can not prove it strictly. Any help?
This is correct. Here's another to see it (assume for simplicity we are working over a field $k$). Let $R_\epsilon : = k[\epsilon]/\epsilon^2$ and let $S_\epsilon:= Spec R_\epsilon.$ This is a scheme with a single $k$-point, $s\in S$ and a one-dimensional tangent space. Now recall that the tangent space can be interpreted as a mapping space (in the category of schemes): $$TG = \{f\in Hom(S_\epsilon, G)\mid f(s) = e\}.$$
On the other hand, $$Hom(G^\vee, \mathbb{G}_a) : = Hom_{Hopf}(k[x], O_G^\vee) \cong Hom_{Hopf}(O_G, k[x]^\vee),$$ where Hom is taken in the category of Hopf algebras. Now $k[x]$ has basis $x^n$ and $k[x]^\vee$ has dual elements $x^\vee_i$ (this is not quite a basis, unless we define the dual more carefully, but that isn't relevant here). Now you can check directly that $$(x_1^\vee)^2 = 0,$$ so given $f\in \hom(G^\vee, \mathbb{G}_a)$, the function $f^\vee: O_G\to k[x]^\vee$ can be restricted to the subring $k[x_1^\vee]\cong R_\epsilon$, giving a tangent vector (in $Hom(Spec R_\epsilon, G)$, of schemes). Conversely, say you have a map $S_\epsilon\to G$ with the closed point going to $e\in G$. Dually, you have a map of co-rings $k\cdot 1^\vee\oplus k\cdot\epsilon^\vee\to O_G^\vee,$ taking $1^\vee$ to the identity. Now remembering the ring structure on $O_G^\vee,$ this extends to a map from the polynomial ring $k[\epsilon^\vee]\to O_G^\vee,$ and setting $x = \epsilon^\vee,$ you can check that this construction is converse to the previous one (in particular, the resulting map is a map of Hopf algebras).
There is a fancy way to generalize this idea in terms of adjoint functors, as follows. Let $Ring_*$ be the category of augmented rings, dual to the category of affine schemes with a basepoint (here rings and Hopf algebras are assumed to be commutative and co-commutative), and let $coRing_*$ be the category of coaugmented corings (with all operations and axioms reversed).
Then the forgetful functor $$\mathtt{forg}:Hopf\to coRing_*$$ (which ignores multiplication but remembers comultiplication and the coaugmetnation — equivalently, unit) has a left adjoint, $$\mathtt{free}:coRing_*\to Hopf.$$ This is actually very easy to see: any coaugmented coring $R$ can be canonically written as $k\cdot 1 \oplus R/k,$ and the free Hopf algebra on $k\oplus R/k$ is simply $k[R/k],$ with comultiplication induced from that on $R$. In particular, plugging in the co-augmented co-ring corresponding to $S_\epsilon,$ which is $R_\epsilon^\vee$, we can compute $\mathtt{free}R_\epsilon^\vee \cong O_{\mathbb{G}_a}$ as Hopf algebras, and the statement you want follows from the universal property.
