Can Dead Weight be sacrificed after it kills a creature?
No. Once a player would get priority, if Dead Weight reduces the toughness of the creature to a value less than 0, or if the damage marked on the creature is equal to or greater than its (new) toughness, State Based Actions will cause the creature to go to the graveyard.

When the creature goes to the graveyard, Dead Weight will follow since it is no longer attached to an object or player.