The formation and dissociation of a Cu-H complex in ZnO is studied by using the density functional theory. We find that a diffusive H atom can be easily trapped by the doped Cu in ZnO, to form a Cu-H complex, and the H atom releasing from the Cu-H complex requires energy of at least 1.5 eV. When an H atom is trapped by the doped Cu in ZnO, this H atom can move around the Cu atom by overcoming an energy barrier of only 0.63 eV. Furthermore, the calculated electronic structures indicate that the H atom in Cu-doped ZnO causes the Cu 3d orbitals to be split and weakens the interaction between Cu 3d and O 2p orbitals, resulting in an upward shift of the occupied defect states. Such an electronic feature is reflected in the optical absorption spectra.