First, make sure you save your game right after the fight starts, if you do happen to die, just reload your savegame and the achievement will still unlock.
The fight with Natla consists of 2 stages, 3 if you count the 'superaction' you need to complete to finish her off.
In the first stage she will be airborne, shooting (3) fireballs at you from a distance, teleporting through the room at random and if she gets close enough, she will try and do a kick attack at Lara.
The trick here is to stay as far away as possible, avoiding the kick attack and at the same time, giving you as much time as possible to dodge her fireball attack. To get past this stage, you will have to target her wings and shoot her down. The target will often automatically lock onto her head, so manually switch it to her wings and shoot both of them until you get the cinematic and Natla is grounded.
In this stage, she basically has only 1 attack, her fireball attack, staying at a distance still is your best bet here as it makes it fairly easy to dodge them or jump out of the way. Natla will have a rage meter now, keep shooting her to fill it up and she will come running at you.
During the adrenaline attack, make sure you summersault over her so that Lara can hit the green target on Natla's back. Dodging to the back or one of the sides will make it hard or even impossible to hit the mark.
Not to worry though, if you miss the green target, Natla will keep charging at you until her rage bar is empty again, giving you plenty of opportunities to hit it. Once you do hit it, Natla will be stunned for a while, giving Lara the chance to keep shooting at the green target, taking away a good chunk of her lifebar. Keep repeating this until it is empty.
Now make sure you hit all 5 'superaction' keys on time, or it will still count as a death during the fight with her and the achievement will not unlock. The keys are: A, B, B, RT and X.