The morphologies, self-assembly structures, and stability of cobalt-phthalocyanines (CoPc) molecules adsorbed on Cu(001) with coverage ranging from 0.2 monolayer (ML) to 1.6 ML are investigated by ultrahigh-vacuum low-temperature scanning tunneling microscopy (UHV LT-STM) at liquid nitrogen temperature. Upon increasing the deposition of CoPc molecules various structures, such as isolated adsorption, quasi-hexagonal structure, root 29 x root 29 structure, are well characterized by the corresponding high-resolution STM images. The CoPc-CoPc intermolecular interaction and CoPc-substrate interfacial interaction dominate the structural evolutions. For the coverage higher than 1 ML, CoPc molecules preferentially locate on top of the molecules underneath and organize into root 58 x root 58 structure. As more and more CoPc molecules adsorb on the first layer, in some root 58 x root 58 regions molecular insertion leads to the formation of the root 29 x root 29 domain to effectively decrease the energy of the whole system.