Showing that $A/J \cong \mathbb C.$
Solution 1:
Apply the first isomorphism theorem for $C^*$-algebras:
Theorem: Let $\varphi: A \to B$ be a $*$-homomorphism between $C^*$-algebras. Then the induced map $$\overline{\varphi}: A/\ker(\varphi) \to \varphi(B): a + \ker (\varphi)\mapsto \varphi(a)$$ is a well-defined $*$-isomorphism.
Apply this to the canonical $*$-homomorphism $$\operatorname{ev}_0: C_0(\mathbb{R})\to \mathbb{C}: f \mapsto f(0).$$