A bit of Tannakian formalism clarifies the situation. Recall that for every abstract group $\Gamma$ there is a notion of "algebraic hull" $\Gamma^{alg}$ constructed as follows: Consider pairs $(\varphi,H)$ where $H$ is an algebraic group over $k$ and $\varphi:\Gamma\to H(k)$ a homomorphism of groups with Zariski-dense image. Maps between such pairs are defined in the obvious was, and one defines $$\Gamma^{alg} := \lim_{(\varphi,H)}H$$ So this is an affine group scheme over $k$. The group $\Gamma^{alg}$ has a universal property, namely that for any algebraic group $H$ over $k$, any map $\Gamma \to H(k)$ factors over $\Gamma^{alg}$. From Tannakian formalism we know:

The canonical map $\Gamma\to\Gamma^{alg}$ induces an equivalnence of monoidal categories between the category of finite dimensional algebraic $\Gamma^{alg}$-representations and finite dimensional $k$-representations of $\Gamma$.

Now, let's apply this to $\Gamma = G(k)$ for a fixed algebraic, not necessarily reductive group over $k$. The universal property of the algebraic hull yields a canonical mmorphism $\Gamma^{alg}\to G$ whose image is the Zariski-closure of $\Gamma$ in $G$. We have then results like this:

Proposition (Deligne, LNM900, p.139): Let $f: H\to G$ be a morphism of affine group schemes over $k$ and let $\omega$ be the induced functor $Rep_k(G) \to Rep_k(H)$ between categories of finite dimensional representations. The morphism $f$ is faithfully flat if and only if $\omega$ is fully faithful and if for every representation $V$ of $G$ every subrepresentation of $\omega(V)$ is isomorphic to the image of a subrepresentation of $V$.

For instance, full flatness of $f$ means also surjectivity of $f$, which is therefore a necessary condition for the induced functor of representations to be fully faithfull - this is what's behind Brian's comment. If you assume $G$ to be reductive then the category $Rep(G)$ is semisimple and the last condition in Deligne's proposition can, maybe, be rephrased in a simpler way.

A bit of Tannakian formalism clarifies the situation. Recall that for every abstract group $\Gamma$ there is a notion of "algebraic hull" $\Gamma^{alg}$ constructed as follows: Consider pairs $(\varphi,H)$ where $H$ is an algebraic group over $k$ and $\varphi:\Gamma\to H(k)$ a homomorphism of groups with Zariski-dense image. Maps between such pairs are defined in the obvious was, and one defines $$\Gamma^{alg} := \lim_{(\varphi,H)}H$$ So this is an affine group scheme over $k$. The group $\Gamma^{alg}$ has a universal property, namely that for any algebraic group $H$ over $k$, any map $\Gamma \to H(k)$ factors over $\Gamma^{alg}$. From Tannakian formalism we know:

The canonical map $\Gamma\to\Gamma^{alg}$ induces an equivalnence of monoidal categories between the category of finite dimensional algebraic $\Gamma^{alg}$-representations and finite dimensional $k$-representations of $\Gamma$.

Now, let's apply this to $\Gamma = G(k)$ for a fixed algebraic, not necessarily reductive group $G$ over $k$. The universal property of the algebraic hull yields a canonical morphism $\Gamma^{alg}\to G$ whose image is the Zariski-closure of $\Gamma$ in $G$. We have then results like this:

Proposition (Deligne, LNM900, p.139): Let $f: H\to G$ be a morphism of affine group schemes over $k$ and let $\omega$ be the induced functor $Rep_k(G) \to Rep_k(H)$ between categories of finite dimensional representations. The morphism $f$ is faithfully flat if and only if $\omega$ is fully faithful and if for every representation $V$ of $G$ every subrepresentation of $\omega(V)$ is isomorphic to the image of a subrepresentation of $V$.

For instance, faithful flatness of $f$ implies surjectivity of $f$, which is therefore a necessary condition for the induced functor of representations to be fully faithful - this is what's behind Brian's comment. If you assume $G$ to be reductive then the category $Rep(G)$ is semisimple (in characteristic zero) and the last condition in Deligne's proposition can, maybe, be rephrased in a simpler way.