To get a battle calculated by the AI, you need to press
during a battle. Pressing this will have the game make decisions for you, which then solves the battle on its own. You won't have to do anything else but press this button, and sometimes tell the game whether or not it can use magic.
It's worth noting that using the AI doesn't necessarily give you a better chance of winning. But whether or not you win or lose the battle doesn't matter for the achievement.
You also don't actually need to finish the battle with the AI, you just need to press the button at least once during the fight. The achievement tracker seems to be working fine, so you can keep track of every successful attempt as well. In addition to this, if you die during combat, reloading the save and repeating the fight will still count.
Once you've had the AI assist you 100 times, the achievement will pop!