Let $X$ be a scheme, for example a curve (so a $1$-dimensional, proper $k$-scheme).
Can anybody explain we the relation between the dualizing sheaf $\omega_C$ on $C$ and a dualizing complex $\omega_C ^{\bullet}$?
Here:https://stacks.math.columbia.edu/tag/0A7A
it is only introduced for rings, but it seems that that's not be a problem to extend it to schemes.
It's the same notion. If you understand how to define the dualizing complex on a scheme, the only observation is that this complex may sometimes be quasi-isomorphic to a single sheaf (this happens when the scheme $X$ is Cohen-Macaulay). So on a CM scheme, it makes sense to call the dualizing complex "the dualizing sheaf." You posted another question recently about Gorenstein rings -- you can probably see why a Gorenstein scheme is one whose dualizing complex is a line bundle.