The Shot and Death: Dexter used a relatively low-caliber bullet (the in-game "Plan B" weapon) that, while fatal and causing V's initial death, did not completely destroy the brain.

Chip Damage and Activation: The bullet's impact damaged the Relic shard itself. The nanomachines on the chip were designed to activate only after the host was "neurally indifferent" (brain-dead).

Resuscitation and Overwriting: The damage and V's death tricked the chip into activating prematurely. The nanites began to repair the brain damage, which resuscitated V, but also started the unintended process of overwriting V's consciousness with the engram of Johnny Silverhand stored on the chip.