Felix Klein, <A HREF="https://link.springer.com/content/pdf/10.1007/BF02297507.pdf">Über die Transformation der elliptischen Funktionen und die Auflösung der Gleichungen fünften Grades</A>, Math. Ann. **14**,‎ 111-172 (1878-1879).

There is also a slightly earlier brief note in Italian, <A HREF="https://archive.org/details/RendicontiIstitutoLombardoS2V10/page/n279">Sull' equazioni dell' Icosaedro nella risoluzione delle equazioni del quinto grado</A>, Rendiconti Reale Istituto Lombardo, Serie 2, vol. 10, p. 253-255‎ (1877).

<IMG SRC="https://ilorentz.org/beenakker/MO/Klein.png" WIDTH="400"/>

Dedekind may have gotten there first, see <A HREF="http://macos.ua.ac.be/~lebruyn/LeBruyn2012a.pdf">Dedekind or Klein</A> (p. 67).

Notation: Klein used capital $J$, there is a factor 1728 difference with the $j$-invariant. (<A HREF="https://math.stackexchange.com/questions/531532/why-1728-in-j-invariant">Read more</A> about the origin of this factor.)