The growth mechanism and chirality formation of a single-walled carbon nanotube (SWNT) on a surface-bound nickel nanocluster are investigated by hybrid reactive molecular dynamics/force-biased Monte Carlo simulations. The validity of the interatomic potential used, the so-called ReaxFF potential, for simulating catalytic SWNT growth is demonstrated. The SWNT growth process was found to be in agreement with previous studies and observed to proceed through a number of distinct steps, viz., the dissolution of carbon in the metallic particle, the surface segregation of carbon with the formation of aggregated carbon clusters on the surface, the formation of graphitic islands that grow into SWNT caps, and finally continued growth of the SWNT. Moreover, it is clearly illustrated in the present study that during the growth process, the carbon network is continuously restructured by a metal-mediated process, thereby healing many topological defects. It is also found that a cap can nucleate and disappear again, which was not observed in previous simulations. Encapsulation of the nanoparticle is observed to be prevented by the carbon network migrating as a whole over the cluster surface. Finally, for the first time, the chirality of the growing SWNT cap is observed to change from (11,0) over (9,3) to (7,7). It is demonstrated that this change in chirality is due to the metal-mediated restructuring process.