The context is in model category, we define the homotopy category of a model category $\operatorname{Ho}\mathcal{C}$ by a functor $$\begin{align} \pi: \mathcal{C} &\to \operatorname{Ho} \mathcal{C} \\\\ X &\mapsto RQX \\\\ f &\mapsto [RQf] \end{align}$$ where $R,Q$ is (co)fibrant replacement. In my lecture note, it's taken to be two seperate steps to define $\operatorname{Ho}\mathcal{C}$. That is, factor $\pi$ into
$$\mathcal{C} \stackrel{\bar{\pi}}{\longrightarrow} \overline{\mathcal{C}} \stackrel{\xi}{\longrightarrow} \mathrm{Ho}(\mathcal{C})$$ Where $\bar{\pi}$ is the identity on objects and given by $f \mapsto [RQf]$ on morphisms, while $\xi$ is given by $X \mapsto RQX$ on objects and the identity on morphisms.
But is it possible if can just take it to be the composition $R\circ Q: \mathcal{C} \to \mathcal{C_{cf}}$ (category of bifibrant objects), then take the localization of $W$?
If you could give an answer in the context of model category or higher category (per Lurie) then that would be very helpful!
You can definitely define $\mathrm{Ho}(\mathcal{C})$ as the localization of $\mathcal{C}_{\mathrm{cf}}$ at the class of (homotopy) equivalences and the canonical functor $\mathcal{C}\to\mathrm{Ho}(\mathcal{C})$ as composite $\mathcal{C}\xrightarrow{RQ}\mathcal{C}_\mathrm{cf}\xrightarrow{\mathrm{localization}}\mathcal{C}_\mathrm{cf}[W^{-1}]$. This is basically just a different way to obtain the same composite as your lecture notes do it, if you model $\mathcal{C}_\mathrm{cf}[W^{-1}]$ by taking homotopy classes of maps in $\mathcal{C}_\mathrm{cf}$ (a priori $\mathcal{C}_\mathrm{cf}[W^{-1}]$ is only defined up to equivalence, and we have different ways to model it; if you only care about $\mathrm{Ho}(\mathcal{C})$ up to equivalence, you may model the localization any way you like).
The reason those lecture notes break it up in two steps is because $\overline{\mathcal{C}}$ itself is also a model for $\mathrm{Ho}(\mathcal{C})$ (as $\xi$ is fully faithful by definition, and essentially surjective because we already have taken homotopy classes of maps), and this model has the advantage that it has the same objects as $\mathcal{C}$, so that we can take the localization functor to act as the identity on objects. This avoids the need to do (in practice very inexplicit) bifibrant replacement of objects, although you still have to do so on morphisms. On the other hand, restricting to bifibrant objects has the advantage that weak equivalences are already homotopy equivalences, and bifibrant objects themselves satisfy quite some pleasant properties.
If you only care about $\mathrm{Ho}(\mathcal{C})$ up to equivalence, you could also just put it to be $\mathcal{C}[W^{-1}]$, but one advantage of the previous approaches is that you actually know the homotopy category is ''at most as large'' as $\mathcal{C}$ is. General localizations of categories can come with mild size issues. Moreover, for model categories we actually have a calculus of fractions that describes the homotopy category, which makes the localization actually tractable (and this was one of the main reasons model categories were introduced: to be able to actually study the localizations of categories at weak equivalences).