The constant-pressure molecular dynamics simulations based on the second-moment approximation of tight-binding scheme have been performed to study the relationship between the resulting crystallization microstructure of the liquid copper and the cooling rate. Below the glass-forming critical cooling rate, the metastable hcp phase and the stable fcc phase can coexist in the resulting configuration with all sorts of proportion and various forms such as layering and phase separation. The sizes and the distributions of the two crystalline phases depend on the cooling rate: the faster the cooling rate the larger percents of the metastable hcp phase and the more easily the layering take place. From the split of peaks of the angular distribution function, for both hcp and fcc phase the faster the cooling rate is the more imperfect they are, which may be considered as the precursor of glass-forming.