To elaborate Andreas Thom's comment into an answer:

Let $X'$ be the set of points of the space $X$ with the discrete topology. As you note, the Stone-Čech compactification $\beta X'$ can be identified with the set of ultrafilters on $X'$ (resp. $X$). Now, since the compactification $\beta X$ is a compact space, the universality of the space of ultrafilters $\beta X'$ tells us that there exists a unique continuous map $f:\beta X' \to \beta X$. Moreover, this map is given by a limit along ultrafilters

$$ f(p) = \lim_{x \to p}\ \iota(x),\qquad p\in\beta X'$$

where $\iota : X' \hookrightarrow X \to \beta X$ is the inclusion. The map is surjective because the space $X$ is dense in $\beta X$, so it's a quotient map.

If you think about it, the explicit description of $f$ as ultrafilter limit makes it clear that it induces your equivalence relation. Put differently, you only need to think about the universal example $C=\beta X$ anyway.