Using the state-of-art dynamical mean-field theory combined with density functional theory method, we have performed systematic study on the temperature and pressure dependent electronic structure of ferromagnetic quantum critical material candidate CeRh$_6$Ge$_4$. At -3.9 GPa and -8.3 GPa, the Ce-4$f$ occupation variation, the local magnetic susceptibility, and the low-frequency electronic self-energy behaviors suggest the Ce-4$f$ electrons are in the localized state; whereas at 6.5 GPa and 13.1 GPa, these quantities indicate the Ce-4$f$ electrons are in the itinerant state. The characteristic temperatures associated with the coherent Kondo screening is gradually suppressed to 0 around 0.8 GPa upon releasing external pressure, indicative of a local quantum critical point. Interestingly, the momentum-resolved spectrum function shows that even at the localized state side, highly anisotropic $mathbf{k}$-dependent hybridization between Ce-4$f$ and conduction electrons is still present along $Gamma$-A, causing hybridization gap in between. The calculations predict 8 Fermi surface sheets at the local-moment side and 6 sheets at the Kondo coherent state. Finally, the self-energy at 0.8 GPa can be well fitted by marginal Fermi-liquid form, giving rise to a linearly temperature dependent resistivity.
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