After my opponent declares a blocking creature, I use an instant to deal 2 damage which kills it. Do they take combat damage to the face?
No, unless the attacking creature has trample.
Once a creature has become blocked, it remains blocked for the rest of combat, no matter what subsequently happens to the creature(s) blocking it. If an attacking creature has all its blockers removed from combat somehow, it will take and deal no combat damage. Only the trample ability allows the attacking creature to deal its damage to the defending player in that case.