Cu-63 nuclear quadrupole resonance (NQR), nuclear antiferromagnetic resonance (AFNMR), and spin-lattice relaxation, as well as Li-7 NMR and relaxation measurements in CuO and in Cu1-xLixO, are used to study the spin dynamics in the paramagnetic and in the antiferromagnetic (AF) phase, and in particular the effects induced by lithium doping. It is argued that Li enters in CuO lattice up to a doping x around 4%; from a comparison of the electric-field gradient (EFG) and of the dipolar field at the nucleus with theoretical estimates, it appears that Li+ is slightly displaced from the Cu lattice site. From NQR, NMR, and AFNMR spectra, information on the Cu hyperfine interaction and the EFG are derived. The Neel temperature T(N) is affected by Li doping to a moderate extent, decreasing from 226 K in CuO to T(N) = 183 K for x = 3.7%. The relaxation rates, driven by the time-dependent part of the magnetic electron-nucleus Hamiltonians, indicate that the Cu2+ spin dynamics is almost unaffected by Li doping for T much greater than T(N) and possibly controlled by valence fluctuations. The Li doping modifies in a dramatic way the spin fluctuations in the AF phase. The relaxation rates in Cu1-xLixO, in fact, increase by orders of magnitude (similarly to the case of copper-oxide precursors of high-T(c) superconductors) and display temperature behaviors quite different from those in CuO. These effects are associated with an effective low-frequency spectral density of the spin fluctuations negligible in CuO, which strongly increases upon Li doping. This increase is related to the fluctuations induced in the local Cu2+ spin configuration as a consequence of the diffusionlike motion of the holes in the AF matrix, implying the insurgence of low-frequency spin excitations. The orders of magnitude of the relaxation rates, their dependences on x and on T, as well as on the measuring frequency, are satisfactorily explained by a picture in which the correlation time tau(h) for the hole-hopping motion is controlled by an energy gap E(x) between localized and itinerant states. Quantitative information on tau(h) and E(x) is thus derived.