This last mission is a one on one battle against Vergil. For the first few seconds, try to learn the pattern of Vergil's attacks. You need to counterattack him after he launches his attacks. The frequencies and speeds of his attacks will only increase as his HP is depleted. Later on, he'll enter devil trigger mode and will summon a double. You'll only have to land quick combos to slowly whittle his HP. Once his HP is critically low, he will kneel and start regenerating his HP while his double will attack in his behalf. Keep killing his double to recover some of your devil trigger gauge. If his HP regenerates enough, he will stop and you'll have to bring it down critically again. SAVE your devil trigger until his HP is critical enough. Once he starts regenerating again, enter devil trigger and attack him to land the finishing blow.
The video below is not outstanding but it should be enough for you to learn Vergil's attacks.