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Dislocation climb is one of the key processes controlling the mechanical response and microstructure evolution of materials at high temperatures. The mechanisms of this phenomenon are still unclear due to its complex nature, which involves interactions between point defects and dislocations. Here we employed large-scale classical molecular-dynamics (MD) simulations to shed light on vacancy-driven dislocation climb in fcc Ni. A Ni crystal with dissociated dislocations was simulated under high-temperature vacancy supersaturation for multiple nanoseconds. A particular advantage of this method is the ability to reveal the underlying mechanisms of dislocation climb without prior knowledge. The study included three different dislocation characters: pure edge, mixed 30, and 60. We found that jog-pair formation initiated with vacancy clustering at partial dislocations, followed by the alignment of vacancies into linear clusters along the <110> direction. This stage was clarified by a detailed analysis of vacancy segregation energies at dissociated dislocations for vacancy clusters of different sizes and configurations. Subsequent stages of jog-pair formation involved local reduction of the stacking-fault region and further jog-pair growth due to vacancy accumulation. From the MD simulations, we traced the full dislocation climb process and compared it with classical descriptions of dislocation climb in fcc metals. Detailed analysis of the MD results allowed us to resolve and describe all stages of dislocation climb from the perspective of dislocation reactions. Based on the changes in dislocation positions, we estimated the climb velocity for the considered dislocations driven by the osmotic force.