Aims. We present the detailed characterisation of a sample of 56 sources serendipitously detected in ALMA band 7 as part of the ALMA Large Program to INvestigate CII at Early Times (ALPINE). These sources, detected in COSMOS and ECDFS, have been used to derive the total infrared luminosity function (LF) and to estimate the cosmic star formation rate density (SFRD) up to z similar or equal to 6.Methods. We looked for counterparts of the ALMA sources in all the available multi-wavelength (from HST to VLA) and photometric redshift catalogues. We also made use of deeper UltraVISTA and Spitzer source lists and maps to identify optically dark sources with no matches in the public catalogues. We used the sources with estimated redshifts to derive the 250 mu m rest-frame and total infrared (8-1000 mu m) LFs from z similar or equal to 0.5 to 6.Results. Our ALMA blind survey (860 mu m flux density range: similar to 0.3-12.5 mJy) allows us to further push the study of the nature and evolution of dusty galaxies at high-z, identifying luminous and massive sources to redshifts and faint luminosities never probed before by any far-infrared surveys. The ALPINE data are the first ones to sample the faint end of the infrared LF, showing little evolution from z similar or equal to 2.5 to z similar or equal to 6, and a "flat" slope up to the highest redshifts (i.e. 4.5<z<6). The SFRD obtained by integrating the luminosity function remains almost constant between z similar or equal to 2 and z similar or equal to 6, and significantly higher than the optical or ultra-violet derivations, showing a significant contribution of dusty galaxies and obscured star formation at high-z. About 14% of all the ALPINE serendipitous continuum sources are found to be optically and near-infrared (near-IR) dark (to a depth Ks similar to 24.9 mag). Six show a counterpart only in the mid-IR and no HST or near-IR identification, while two are detected as [C II] emitters at z similar or equal to 5. The six HST+near-IR dark galaxies with mid-IR counterparts are found to contribute about 17% of the total SFRD at z similar or equal to 5 and to dominate the high-mass end of the stellar mass function at z> 3.