The deposition of thin uniform dielectric layers on graphene is important for its successful integration into electronic devices. We report on the atomic layer deposition (ALD) of Al2O3 nanofilms onto graphene grown by chemical vapor deposition onto copper foil. A pretreatment with deionized water (DI H2O) for graphene functionalization was carried out, and, subsequently, trimethylaluminum and DI H2O were used as precursors for the Al2O3 deposition process. The proper temperature regime for this process was adjusted by means of the ALD temperature window for Al2O3 deposition onto a Si substrate. The obtained Al2O3/graphene heterostructures were characterized by Raman and X-ray photoelectron spectroscopy, ellipsometry and atomic force and scanning electron microscopy. Samples of these heterostructures were transferred onto glass substrates by standard methods, with the Al2O3 coating serving as a protective layer during the transfer. Raman monitoring at every stage of the sample preparation and after the transfer enabled us to characterize the influence of the Al2O3 coating on the graphene film.