Let $\tilde C$ be the left Bousfield localization of $C$. As categories $C=\tilde C$, but $\tilde C$ has more weak equivalences. In particular, the identity functor induces a functor $\varphi\colon \operatorname{Ho}(C)\rightarrow \operatorname{Ho}(\tilde C)$. Let $\mathcal L=\ker \varphi$, i.e. $\mathcal L\subset \operatorname{Ho}(C)$ is the full subcategory spanned by the objects which become trivial in $\operatorname{Ho}(\tilde C)$. If $\tilde C$ is stable and $\varphi$ preserves homotopy colimits (and these are the suitable conditions, I think) then $\varphi$ is an exact functor between triangulated categories, so $\mathcal L$ is a thick subcategory of $\operatorname{Ho}(C)$ and $\varphi$ induces a functor from the Verdier quotient $\bar\varphi\colon\operatorname{Ho}(C)/\mathcal L\rightarrow \operatorname{Ho}(\tilde C)$. Let us show that $\bar\varphi$ is an equivalence of categories. It is enough to prove that the canonical composition of 'projection' functors $\tilde C=C\rightarrow\operatorname{Ho}(C)\rightarrow \operatorname{Ho}(C)/\mathcal L$ satisfies the universal property of $\tilde C\rightarrow \operatorname{Ho}(\tilde C)$. Suppose $\psi\colon C\rightarrow D$ is a functor which sends all weak equivalences in $\tilde C$ to isomorphisms. Since weak equivalences in $C$ are weak equivalences in $\tilde C$, $\psi$ factors through $C\rightarrow \operatorname{Ho}(C)$ in an essentially unique way. Let $\bar\psi\colon \operatorname{Ho}(C)\rightarrow D$ be the factorizaton. Recall that $\operatorname{Ho}(C)/\mathcal L$ is the localization of $\operatorname{Ho}(C)$ inverting those maps whose mapping cone is in $\mathcal L$. Any such map is represented by a zig-zag of weak equivalences in $\tilde C$. Since $\psi$ sends weak equivalences in $\tilde C$ to isomorphisms then $\bar\psi$ factors through $\operatorname{Ho}(C)\rightarrow \operatorname{Ho}(C)/\mathcal L$ in an essentially unique way. In particular $\psi$ factors through the composite $\tilde C\rightarrow \operatorname{Ho}(C)/\mathcal L$. The essential uniqueness of this factorization follows from the aforementioned essential uniqueness of the two intermediate steps. I hope this argument is correct and convincing, and that I haven't missed any necessary hypothesis.