A graph argument settles this issue very nicely, as follows.

Consider a linear algebraic group $G$ over a field $k$ of characteristic 0, and let $\mathfrak{g}$ be its Lie algebra.  For a finite-dimensional $k$-vector space $V$, let $f:\mathfrak{g} \rightarrow \mathfrak{gl}(V)$ be a representation.  Provided that $G$ is connected, since ${\rm{char}}(k) = 0$ clearly there is at most one $k$-homomorphism $\rho:G \rightarrow {\rm{GL}}(V)$ such that ${\rm{Lie}}(\rho) = f$, so assuming $G$ is connected we seek conditions under which such a $\rho$ always exists.

Let $\mathfrak{h} \subset \mathfrak{g} \times \mathfrak{gl}(V)$ be the graph of $f$.  Clearly if $\rho$ is to exist and  the $k$-subgroup $H \subset G \times {\rm{GL}}(V)$ is its graph then $H$ is connected and $\mathfrak{h} = {\rm{Lie}}(H)$, so $H$ is uniquely determined (as a $k$-subgroup of $G \times {\rm{GL}}(V)$) since ${\rm{char}}(k) = 0$.  So we seek conditions under which the Lie subalgebra $\mathfrak{h}$ of $\mathfrak{g} \times \mathfrak{gl}(V)$ "exponentiates" to a connected closed $k$-subgroup $H$.

Assume $G$ is its own derived group, so $\mathfrak{g}$ is its own derived subalgebra and hence the same for $\mathfrak{h}$. It is a general fact over fields of char. 0 (see Cor. 7.9 in Ch. II of Borel's textbook on algebraic groups) that the derived subalgebra of any Lie subalgebra of a linear algebraic group "exponentiates" to a connected closed $k$-subgroup.  So $\mathfrak{h} = {\rm{Lie}}(H)$ for a unique connected closed $k$-subgroup $H \subset G \times {\rm{GL}}(V)$.  The necessary and sufficient condition for $f$ to arise from some $\rho$ is that ${\rm{pr}}_1:H \rightarrow G$ is an isomorphism.  

This projection has Lie algebra map $\mathfrak{h} \rightarrow \mathfrak{g}$ that is the analogous projection which is visibly an isomorphism (due to the definition of $\mathfrak{h}$ as the graph of $f$), so $H \rightarrow G$ is an isogeny.  As such, its kernel is etale (since ${\rm{char}}(k) = 0$) and hence central (since $H$ is connected), so it is a finite central $k$-subgroup of $H$. Thus, we just need that $G$ admits no nontrivial isogenous (smooth) connected central extension.  This is automatic when $G$ is assumed to be semisimple and simply connected (in the sense of algebraic groups).

So we win whenever $G$ is a connected semisimple $k$-group that is simply connected.