Kaervek, le châtieur

You cast the copy while the ability is resolving and still on the stack. You can’t wait to cast it later in the turn.
A player can commit only one crime per spell or ability they control. Targeting multiple opponents, permanents, spells, abilities, and/or cards with the same spell or ability doesn’t constitute committing multiple crimes.
A player commits a crime as they cast a spell, activate an ability, or put a triggered ability on the stack that targets at least one opponent, at least one permanent, spell, or ability an opponent controls, and/or at least one card in an opponent’s graveyard.
The spell or ability that constituted a crime doesn’t have to have resolved yet or at all. As soon as you’re finished casting the spell, activating the ability, or putting the triggered ability on the stack, you’ve committed a crime.
If you don’t want to cast the copy, you can choose not to; the copy ceases to exist the next time state-based actions are performed.
Because you’re paying the spell’s costs, if the spell has in its mana cost, you may choose its value as normal.
The copy is created in and cast from exile, so effects such as that of Aven Interrupter’s last ability will apply.
For example, an ability that triggers when you cast a spell that targets an opponent will trigger at the same time as an ability that triggers whenever you commit a crime. Those abilities can be put on the stack in either order (if you control them both), and they’ll both resolve before the spell that caused them to trigger.
Changing the target or targets of a spell or ability won’t affect whether or not the controller of that spell or ability has committed a crime. Only the initial targets chosen for that spell or ability are used to determine whether or not its controller committed a crime.