We study the characteristics of chaos evolution of initially localized energy excitations in the one-dimensional nonlinear disordered Klein-Gordon lattice of anharmonic oscillators, by computing the time variation of the fundamental frequencies of the motion of each oscillator. We focus our attention on the dynamics of the so-called "weak chaos"and "strong chaos"spreading regimes [Laptyeva et al., 2010], for which Anderson localization is destroyed, as the initially restricted excitation at the central region of the lattice propagates in time to more lattice sites. Based on the fact that large variations of the fundamental frequencies denote strong chaotic behavior, we show that in both regimes chaos is more intense at the central regions of the wave packet, where also the energy content is higher. On the other hand, the oscillators at the wave packet's edges, through which the energy propagation happens, exhibit regular motion up until the time they gain enough energy to become part of the highly excited portion of the wave packet. Eventually, the percentage of chaotic oscillators remains practically constant, despite the fact that the number of excited sites grows as the wave packet spreads, but the portion of highly chaotic sites decreases in time. Thus, albeit the number of chaotic oscillators is constantly growing the strength of their chaotic behavior decreases, indicating that although chaos persists it is becoming weaker in time. We show that the extent of the zones of regular motion at the edges of the wave packet in the strong chaos regime is much smaller than in the weak chaos case. Furthermore, we find that in the strong chaos regime the chaotic component of the wave packet is not only more extended than in the weak chaos one, but in addition the fraction of strongly chaotic oscillators is much higher. Another important difference between the weak and strong chaos regimes is that in the latter case a significantly larger number of frequencies is excited, even from the first stages of the evolution. Moreover, our computations confirmed the shifting of fundamental frequencies outside the normal mode frequency band of the linear system in the case of the so-called "selftrapping"regime where a large part of the wave packet remains localized.