I need some references for the definition and basic properties for divisor of one-forme $\omega\in\Omega_{K(\mathcal{C})/K}$ with $\mathcal{C}$ a (integral? smooth?) projective curve over a (algebraically closed/perfect?) field $K$. I guess it's come from the fact that $\Omega_{X/k}$ is locally free of rank $1=\operatorname{dim}(\mathcal{C})$ and there is a correspondence between invertible sheaves and divisors but I guess there is some more direct computations by taking (as on Riemann surfaces) the valuations $v_P\left(\frac{\omega}{dt}\right)$ with $t$ a uniformizer of $\mathcal{O}_{P}$.
I don't see it in Hartshorne, nor Liu, nor Görtz-Wedhorn volume I) but maybe (surely) I missed it. Thanks!
One reference is section II.4 of Silverman's The Arithmetic of Elliptic Curves. I think Proposition 4.3 in particular covers what you're looking for. However, he omits some proofs, referring the reader to Chapter IV of Hartshorne and Chapter 3 of Shafarevich's Basic Algebraic Geometry, vol. 1.
I want to add that you're not so far away from the correspondence between invertible sheaves and divisors. One direction is: given an invertible sheaf $\newcommand{\L}{\mathscr{L}} \newcommand{\O}{\mathscr{O}} \mathscr{L}$, we take a nonzero rational section $s$ and compute its divisor. To compute $\renewcommand{\div}{\operatorname{div}} \div(s)$, we take an open cover trivializing $\L$ (i.e., such that $\L|_U \cong \O_X|_U$ for each $U$) on which $s$ corresponds to a rational function $f$. Then $v_P(s) = v_P(f)$ for any point $P \in U$. That's what you've done: given a differential $\omega$ (i.e., a section of $\Omega_{X/k}$), on a neighborhood $U$ of $P$ with local coordinate $t$ we can write $\omega = f(t)\, dt$ for some rational function $f$. Then the isomorphism $\Omega_{X/k}|_U \cong \O_X|_U$ is just deleting the $dt$, so $v_P(\omega) = v_P(f)$.