Let $G$ be a connected  reductive group over an algebraically closed field $k$.
Write $G^{\rm ss}=[G,G]$ (which is semisimple)
and let $G^{\rm sc}$ denote the universal cover of the semisimple group $G^{\rm ss}$;
then $G^{\rm sc}$ is simply connected.
Consider the composite homomorphism
$$\rho\colon G^{\rm sc}\twoheadrightarrow G^{\rm sc}\hookrightarrow G.$$

Let $T\subseteq G$ be a maximal torus. Consider the maximal tori $T^{\rm ss}=T\cap G^{\rm ss}\subset G^{\rm ss}$, $T^{\rm sc}=\rho^{-1}(T)\subset G^{\rm sc}$.
We write $X^*$ for the character group, and $X_*$ for the cocharacter group.
Clearly,
\begin{multline*}
X^*(G)\cong\{x\in X^*(T)\ |\ x|_{T^{\rm ss}}=1\}\\
=\{x\in X^*(T)\ |\ \langle x,u\rangle=0\ \ 
\forall u\in X_*(T^{\rm ss})\}  
=:X_*(T^{\rm ss})^\bot
\end{multline*}

> **Proposition.** $X^*(G)\cong X^*(T)^W$.

*Proof.* Write $X=X^*(T)$, $X^\vee=X_*(T)$.
Let $R=R(G,T)\subset X$ denote the root system, 
and $R^\vee=R^\vee(G,T)\subset X^\vee$ denote the coroot system.
Let $B\subset G$ be a Borel subgroup containing $T$.
Let $S=S(G,T,B)\subset R\subset X$ denote the system of simple roots, 
and $S^\vee=S^\vee(G,T,B)\subset R^\vee\subset  X^\vee$ denote the system of simple coroots. 
Then $S^\vee$ is a basis of $X_*(T^{\rm sc})$ embedded in $X$.

Since the homomorphism $T^{\rm sc}\to T^{\rm ss}$ is surjective with finite kernel,
the induced homomorphism
$X_*(T^{\rm sc})\to X_*(T^{\rm ss})$ is injective with finite cokernel.
We conclude that 
$$ X^*(G)\cong X_*(T^{\rm ss})^\bot =X_*(T^{\rm sc})^\bot= (S^\vee)^\bot.$$

We know that the Weyl group $W=W(G,T)$ is generated by the reflections $s_\alpha$ for $\alpha\in S$.
These reflections act on $X=X^*(T)$ by
$$s_\alpha(x)= x-\langle x,\alpha^\vee\rangle \alpha\quad\ \text{for}\ \, x\in X, \,\alpha\in S;$$
see, for instance, Subsection 1.1 in Tonny A. Springer,  Reductive groups. Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 3–27.
We write $X^W$ for the subgroup of $W$-fixed points of $X$; then
\begin{multline*}
 X^W=\{x\in X\ |\ s_\alpha(x)=x\ \ \forall \alpha\in S\}\\
=\{x\in X\ |\ \langle x,\alpha^\vee\rangle =0 \ \ \forall \alpha^\vee\in S^\vee\}= (S^\vee)^\bot.
\end{multline*}
Thus  $X^*(G)\cong X_*(T^{\rm ss})^\bot = (S^\vee)^\bot=X^W$, as required.