We explore the formation process of a black hole ( BH ) through the pair-instability collapse of a rotating Population III very massive star in axisymmetric numerical relativity . As the initial condition , we employ a progenitor star which is obtained by evolving a rapidly rotating zero-age main sequence ( ZAMS ) star with mass 320 M _ { \odot } until it reaches a pair instability region . We find that for such rapidly rotating model , a fraction of the mass , \sim 10 M _ { \odot } , forms a torus surrounding the remnant BH of mass \sim 130 M _ { \odot } and an outflow is driven by a hydrodynamical effect . We also perform simulations , artificially reducing the initial angular velocity of the progenitor star , and find that only a small or no torus is formed and no outflow is driven . We discuss the possible evolution scenario of the remnant torus for the rapidly rotating model by considering the viscous and recombination effects and show that if the energy of \sim 10 ^ { 52 } erg is injected from the torus to the envelope , the luminosity and timescale of the explosion could be of the orders of 10 ^ { 43 } erg/s and yrs , respectively . We also point out the possibility for observing gravitational waves associated with the BH formation for the rapidly rotating model by ground-based gravitational-wave detectors .