I am trying to figure out if the Executioner assassin (Heroes of Shadow) applies its dexterity to its melee basic attacks. The Attack Finesse Class feature says: You can use Dexterity instead of Strength for your melee basic attacks. Does this mean substitute dex only for the attack roll and use strength for the damage roll as normal, or use dexterity for both attack and damage?
Both attack and damage
Where it says "use X" for your attacks, replace all instances of the original stat with X. In the cases where it is specifically for attack bonus (like melee training) it'll explicitly call out the difference.
This is an understandable error, especially when you look at the feat melee training.The feat melee training allows you to sub STR with a different stat on the attack roll but you may only use half of the subbed stats value on the damage roll. The exception with attack finesse is that it states: "You can use Dexterity instead of Strength for your melee basic attacks" and not "You can use Dexterity instead of Strength for your melee basic attack rolls"
So for all basic melee attacks (and thus attacks of opportunity triggered when an enemy leaves a combat space adjacent to you) you may use your DEX instead of your STR for the entire action.
In short; Attack Finesse allows Assassins to use their DEX instead of their STR for both basic melee attack and damage rolls.