  We present the results of $^{75}$As nuclear magnetic resonance (NMR), nuclear
quadrupole resonance (NQR), and resistivity measurements in KFe$_2$As$_2$ under
pressure ($p$). The temperature dependence of the NMR shift, nuclear
spin-lattice relaxation time ($T_1$) and resistivity show a crossover between a
high-temperature incoherent, local-moment behavior and a low-temperature
coherent behavior at a crossover temperature ($T^*$). $T^*$ is found to
increase monotonically with pressure, consistent with increasing hybridization
between localized $3d$ orbital-derived bands with the itinerant electron bands.
No anomaly in $T^*$ is seen at the critical pressure $p_{\rm c}=1.8$ GPa where
a change of slope of the superconducting (SC) transition temperature $T_{\rm
c}(p)$ has been observed. In contrast, $T_{\rm c}(p)$ seems to correlate with
antiferromagnetic spin fluctuations in the normal state as measured by the NQR
$1/T_1$ data, although such a correlation cannot be seen in the replacement
effects of A in the AFe$_2$As$_2$ (A= K, Rb, Cs) family. In the superconducting
state, two $T_1$ components are observed at low temperatures, suggesting the
existence of two distinct local electronic environments. The temperature
dependence of the short $T_{\rm 1s}$ indicates nearly gapless state below
$T_{\rm c}$. On the other hand, the temperature dependence of the long
component 1/$T_{\rm 1L}$ implies a large reduction in the density of states at
the Fermi level due to the SC gap formation. These results suggest a real-space
modulation of the local SC gap structure in KFe$_2$As$_2$ under pressure.
