The Lyman-{alpha} line in the ultraviolet (UV) and the [CII] line in the far-infrared (FIR) are widely used tools to identify galaxies in the early Universe and to obtain insights into interstellar medium (ISM) properties in high-redshift galaxies. By combining data obtained with ALMA in band 7 at ~320GHz as part of the ALMA Large Program to INvestigate [CII] at Early Times (ALPINE) with spectroscopic data from DEIMOS at the Keck Observatory, VIMOS and FORS2 at the Very Large Telescope, we assembled a unique sample of 53 main-sequence star-forming galaxies at 4.4<z<6 in which we detect both the Lyman-{alpha} line in the UV and the [CII] line in the FIR. The goal of this paper is to constrain the properties of the Ly{alpha} emission in these galaxies in relation to other properties of the ISM. We used [CII], observed with ALMA, as a tracer of the systemic velocity of the galaxies, and we exploited the available optical spectroscopy to obtain the Ly{alpha}-[CII] and ISM-[CII] velocity offsets. We find that 90% of the selected objects have Ly{alpha}-[CII] velocity offsets in the range 0<{Delta}v_Ly{alpha}-[CII]_<400km/s, in line with the few measurements available so far in the early Universe, and significantly smaller than those observed at lower redshifts. At the same time, we observe ISM-[CII] offsets in the range -500<{Delta}v_ISM-[CII]_<0km/s, in line with values at all redshifts, which we interpret as evidence for outflows in these galaxies. We find significant anticorrelations between {Delta}v_Ly{alpha}-[CII]_ and the Ly{alpha} rest-frame equivalent width EW_0_(Ly{alpha}) (or equivalently, the Ly{alpha} escape fraction f_esc_(Ly{alpha})): galaxies that show smaller {Delta}v_Ly{alpha}-[CII]_ have larger EW_0_(Ly{alpha}) and f_esc_(Ly{alpha}). We interpret these results in the framework of available models for the radiative transfer of Ly{alpha} photons. According to the models, the escape of Ly{alpha} photons would be favored in galaxies with high outflow velocities, producing large EW_0_(Ly{alpha}) and small {Delta}v_Ly{alpha}-[CII]_, in agreement with our observations. The uniform shell model would also predict that the Ly{alpha} escape in galaxies with slow outflows (0<v_out_<300km/s) is mainly determined by the neutral hydrogen column density (NHI) along the line of sight, while the alternative model by Steidel et al. (2010ApJ...717..289S) would more highly favor a combination of NHI at the systemic velocity and covering fraction as driver of the Ly{alpha} escape. We suggest that the increase in Ly{alpha} escape that is observed in the literature between z~2 and z~6 is not due to a higher incidence of fast outflows at high redshift, but rather to a decrease in average NHI along the line of sight, or alternatively, a decrease in HI covering fraction.