Landing a killing blow in V.A.T.S. will immediately restore all of your Action Points, allowing the player to begin a new series of attacks immediately. However, foes killed by allies (or other means) will not trigger the perk, even while in V.A.T.S.
In Fallout: New Vegas, the perk only gives back 20 AP on a kill in V.A.T.S. Although this is a significant reduction to the effectiveness of the perk, the perk is still very useful with high power, low AP cost weapons. The effectiveness of Grim Reaper's Sprint is dependent upon three things: AP cost of the weapon, damage of the weapon, and toughness of the enemy.
If a companion, the Mysterious Stranger, a car explosion, or the explosion caused by targeting an enemy's grenade kills the target, your AP will not be restored. This also happens if the enemy is killed by splash damage or the Tesla cannon's electrical field.
Knocking an essential non-player character unconscious counts as "killing" them for the purposes of this perk.
In Fallout 3, the G.E.C.K. script actually gives the player 1000 AP after a kill, rather than restoring the player's exact amount of AP.
Be aware that in Fallout: New Vegas, some weapons will not gain another shot in V.A.T.S from only 20 AP. Even in this case, the perk retains some usefulness in that it does allow you to go back into V.A.T.S sooner and fire again, even if you cannot immediately return to V.A.T.S mode and fire again.