To earn "First Strike", you must first play through the story until 'The Sunken City'. At the end of this region there is a small tutorial.
From this point on, it can be earnt in any fight.
To perform a fatal strike, you must first stun the enemy. Different types of enemy are weaker to certain types of attack (magic/physical etc). These are indicated by the colours Red/Green/Blue.
If you press RB after scanning the enemy, you can see how close to being stunned they are. Just before the bar is about to empty, a warning glyph will flash on the enemy. This is not the cue, just a warning that its close.
When the coloured bar is empty, a coloured glyph will appear over the enemy for a split second. It is in this split second you have to hit RT. If timed correctly, you will perform a fatal strike and unlock the achievement.
Two points to note -
* Despite what the other guide says, you do not need to be in over-limit. That is for burst artes, unlocked later in the game.
** You do not need to "kill" the enemy, or at least I didn't. I unlocked it against one of the Tweedles in a later tutorial, mid fight. As this is a boss who can't be killed in one shot, maybe the achievement unlocks without a death.
Any questions, just ask.