Formation process of single-walled carbon nanotubes (SWNTs) by the catalytic chemical vapor deposition method is studied by molecular dynamics simulations. The random deposition process of carbon atoms to a nano-scale metal catalyst is realized with the multi-body potentials based on density functional theory calculations of small metal-carbon and metal-metal clusters A nanotube cap structure is generated when pieces of the hexagonal network structure emerges from the metal-carbon binary cluster and then coalesces. Furthermore, the tendency of graphitization of carbon atoms is compared for Ni, Co, Fe metal clusters. The Co cluster has a pseudo crystal structure where metal atoms are regularly allocated and embedded in the hexagonal carbon network. On the other hand, carbon atoms cover the entire surface of the Fe cluster. This implies stronger interaction between the graphitic lattice and Co atoms, hence higher graphitization ability. Finally, the mechanism of chirality determination by the stability of nanotube cap structure is discussed in order to explain the recent experimental finding that the near-armchair nanotubes are more dominantly produced for small diameter nanotubes.