For staking a vampire through the heart in Pathfinder, what type of DC value or combat method should be used in order to perform the staking?
The description of the Vampire says you can stake one if its helpless.
The rules are optional, but Ultimate Combat does have rules for Called Shots. According to those rules, hitting the heart is a -10 to an attack. Staking a vamp is specifically listed, and requires either a Crit or damage equal to half the vampire's hit points (min 50...), which forces a Fort save or be staked.