Let $G \subset SL(n,\mathbb{C})$ be a cyclic subgroup of finite order, Is it true that $\mathbb{C}^n /G$ is toric ? If not then when it is ?

Yes, it is true. Let $G \cong \mathbb{Z}/m$ act by $\mathrm{diag}(\zeta^{a_1}, \zeta^{a_2}, \ldots, \zeta^{a_n})$, where $\zeta$ is a primitive $n$th root of unity. Let $S$ be the semigroup $\{ (b_1, \ldots, b_n) \in \mathbb{Z}_{\geq 0}^n : \sum a_i b_i \equiv 0 \mod m \}$. Then the quotient is Spec of the semigroup ring $\mathbb{C}[S]$. Since the semigroup is torsion free, finitely generated and saturated, the corresponding Spec is an affine toric variety. 

