You can gain progress on this achievement by simply activating Dragonshift at any time during the battle, and then winning the battle. You don't have to stay in Dragonshift or get the final hit in dragon form. So if you want you can farm this on early game enemy encounters by activating Dragonshift right at the start, turning it off again, and then finishing the battle. Put it on Casual Mode for quicker battles.
You get the Dragonshift ability as part of the story right near the start and must be playing as Yuma to activate it. Hold LB then press RB to activate, and do the same to turn it off.
Please note that all achievement trackers in this game only show progress when you save your game, however they achievements DO track properly behind the scenes and will unlock immediately when you complete the requirements.