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, $$ X^*(G)\cong\{x\in X^*(T)\ |\ x|_{T^{\rm ss}}=1\} =\{x\in X^*(T)\ |\ \langle x,u\rangle=0\ \ \forall u\in X_*(T^{\rm ss})\} =:X_*(T^{\rm ss})^\bot $$

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.

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.