Growth of Fe nanostripes on a vicinal Cu(111) surface is investigated on the atomic scale by performing molecular dynamics and kinetic Monte Carlo simulations. We involve in our study the kinetic mechanisms of atomic incorporation recently reported by Mo et al. [Phys. Rev. Lett. 94, 155503 (2005)]. The atomistic processes responsible for the interlayer mass transport and the formation of Fe stripes of 1 ML height are identified. We demonstrate that strain relaxations at steps have a strong impact on the self-assembly of one-dimensional Fe atomic structures on vicinal Cu(111).