Dragons Dogma Dark Arisen

Dragonforging is a method of upgrading your weapons ans armor beyond three-star rating. In order to dragonforge equipment, you must defeat a dragon or dragonkin.

The only dragonkin you can fight during the main story is a Drake, which can be found in Devilfire Grove, just south of the Shadow Fort.

After defeating the Dragon in the main story, a post-game will start, which will affect the world in a few ways, mostly by changing enemy spawns. At this point, plenty of Drakes, Wyverns, Wyrms, and Dragons can be found to use for dragonforging. Defeating the Ur-Dragon can also dragonforge your gear.

Killing the Dragon during the quest 'The Final Battle' will dragonforge ALL the gear you have equipped automatically, effectively making this achievement unmissable.

For any other dragonkin fight, each piece of equipment you have equipped has a small chance to be dragonforged, with higher-upgraded equipment more likely to receive the upgrade. More powerful dragonkin have a higher chance of dragonforging your gear, as well. Only gear equipped on your Arisen can be dragonforged, but you can give dragonforged gear to your pawn aftwrward.

Also note that you will sometimes simply recieve dragonforged gear for no apparent reason, particularly when near Devilfire Grove or a post-game dragonkin spawn. This is due to the dragonkin being killed by the environment, such as accidentally wandering into a lake or the ocean.

EDIT: There are also some dragonkin in Bitterblack Isle, the DLC dungeon. While this is a postgame area with difficulty tuned to characters of level 100-ish, it can be accessed prior to defeating the Dragon, and as such is another possible early source of dragonforged gear, as well as the Devilfire Grove drake.