If $(X,d)$ is compact metric, $C(X)$ in the sup norm is a Banach space.
Fix $p \in X$.
For $x \in X$ define $f_x: X \to \Bbb R$ by $f_x(y)=d(y,x)-d(y,p)$. This $f_x$ is well-defined and continuous, as the metric is a continuous function.
Now check that $F: X \to C(X)$ defined by $F(x)=f_x$ is an isometric embedding.
So $X$ embeds isometrically into $C(X)$, and a classical fact is that $X$ (being compact metric) is a continuous image of the Cantor set $2^\omega$, so we have $\phi: 2^\omega \to X$ a continuous surjection and this induces an isometric injection $\phi^\ast: C(X) \to C(2^\omega): \phi^\ast(f)=f \circ \phi$, another classic fact.
Moreover $C(2^\omega)$ embeds isometrically into $C([0,1])$ by linear interpolation, essentially. This is also well-known.
So we can combine $$X \hookrightarrow C(X) \hookrightarrow C(2^\omega) \hookrightarrow C([0,1])$$
as a chain of isometric embeddings.