The Lyapunov spectrum of a homoclinic class is defined as the closure of the Lyapunov spectra of its periodic points. We study this spectrum in the $C^1$-generic setting. For homoclinci classes, we prove that every interior value of the spectrum is realized by an ergodic invariant measure and derive further properties of the corresponding Lyapunov level sets. Our approach is based on a class of zero-entropy ergodic measures called GIKN.