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Helium atom beam scattering has been used to study the growth of Cu films on Cu( 001) surfaces for a wide range of temperatures and coverages. It is shown that this technique provides a powerful means for a detailed structural characterization during growth of the film. Useful information on decisive kinetic parameters, such as the migration coefficient of adatoms, are obtained. The evolving, rough growth front for deposition at low substrate temperatures is analyzed in the light of the dynamical scaling hypothesis. A growth instability is observed, which might be attributed to an asymmetry in the vertical and horizontal mobility of deposited atoms, associated with an excess energy barrier at step edges.