All of the creatures will deal their respective damage at the same time. Since Sengir Vampire's +1/+1 ability is a triggered ability, it uses the stack, and before any of the +1/+1 counters go onto the Vampire, it will die.
State-based actions (like death from damage) will happen before the triggered ability on the stack can resolve.
If any of the creatures involved have first strike, however, this changes things:
- If the Sengir Vamire has first strike, it will deal it's damage before the blockers, killing them before they have a chance to deal damage, giving it the +1/+1 counters
- If the blockers have first strike, they deal damage to the vampire before it deals damage to them, killing the vampire while they remain alive
- If all parties have first strike, the outcome is the same as if none of them did.