We present infrared spectra (0.1-1 eV) of electrostatically gated bilayer graphene as a function of doping and compare it with tight binding calculations. All major spectral features corresponding to the expected interband transitions are identified in the spectra: a strong peak due to transitions between parallel split-off bands and two onset-like features due to transitions between valence and conduction bands. A strong gate voltage dependence of these structures and a significant electron-hole asymmetry is observed that we use to extract several band parameters. Surprisingly, the structures related to the gate-induced bandgap are much less pronounced in the experiment than predicted by the tight binding model.