We present a theoretical analysis of the capacitance of a double quantum dot in the charge and spin qubit configurations probed at high-frequencies. We find that, in general, the total capacitance of the system consists of two state-dependent terms: The quantum capacitance arising from adiabatic charge motion and the tunnelling capacitance that appears when repopulation occurs at a rate comparable or faster than the probing frequency. The analysis of the capacitance lineshape as a function of externally controllable variables offers a way to characterize the qubits' charge and spin state as well as relevant system parameters such as charge and spin relaxation times, tunnel coupling, electron temperature and electron g-factor. Overall, our analysis provides a formalism to understand dispersive qubit-resonator interactions which can be applied to high-sensitivity and non-invasive quantum-state readout.