If Vein Drinker's first ability causes Vein Drinker and the targeted creature to deal lethal damage to each other, they'll be destroyed at the same time. Vein Drinker's triggered ability will trigger, but it will have been put into a graveyard before it would receive any counters. The ability will do nothing when it resolves.

Each time a creature is put into a graveyard from the battlefield, check whether Vein Drinker had dealt damage to it at any time during that turn. If so, Vein Drinker's third ability will trigger. It doesn't matter whether the damage was combat damage or not. It also doesn't matter who controlled the creature or whose graveyard it was put into.

When the first ability resolves, if the targeted creature is still on the battlefield but Vein Drinker is not, Vein Drinker will still deal damage equal to its power as it last existed on the battlefield to the targeted creature.

When the first ability resolves, if the targeted creature is no longer on the battlefield, the ability doesn't resolve for having no legal targets. No damage is dealt.