Landing a killing blow in V.A.T.S. will immediately restore all Action Points, allowing the beginning of 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.
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 1000 AP after a kill, rather than restoring the 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 going back into V.A.T.S sooner and fire again, even if one cannot immediately return to V.A.T.S and fire again.