See V.L. Popov, E.B. Vinberg, Invariant Theory, Encyclopaedia of Mathematical Sciences, Vol. 55, Algebraic Geometry IV, Springer-Verlag, Berlin, 1994.
You can find the invariants of a binary quartic in 0.12 (p. 141). Fixing values of the two basic invariants, you obtain a closed set, which contains only finitely many orbits (Theorem 8.8, p.247) and only one closed orbit (Corollary of Theorem 4.7, p.189). Within such a closed set, the classification is given by means of multilicities multiplicities of roots, as in Popov's note cited by Bart.