We discuss the detection of 14 rovibrational lines of CH$^+$, obtained toward the planetary nebula NGC 7027 with the iSHELL spectrograph on NASA's Infrared Telescope Facility (IRTF) on Maunakea. Our observations in the 3.49-4.13 μm spectral region, obtained with a 0"375 slit width that provided a spectral resolving power λ/Δλ ~ 80,000, have resulted in the unequivocal detection of the R(0)-R(3) and P(1)-P(10) transitions within the v = 1-0 band of CH$^+$. The R-branch transitions are anomalously weak relative to the P-branch transitions, a behavior that is explained accurately by rovibronic calculations of the transition dipole moment reported in a companion paper. Nine infrared transitions of H₂ were also detected in these observations, comprising the S(8), S(9), S(13), and S(15) pure rotational lines; the v = 1-0 O(4)-O(7) lines; and the v = 2-1 O(5) line. We present a photodissociation model, constrained by the CH$^+$ and H₂ line fluxes that we measured, that includes a detailed treatment of the excitation of CH$^+$ by inelastic collisions, optical pumping, and chemical ("formation") pumping. The latter process is found to dominate the excitation of the observed rovibrational lines of CH$^+$, and the model is remarkably successful in explaining both the absolute and relative strengths of the CH$^+$ and H₂ lines.