WARNING MAY GLITCH!!!
Some people have killed the 10 dragons and the achievement didn't unlock. To fix this, save before your last encounter with the final dragon. It didn't unlock for me the first time i killed 10 dragons, so i loaded my save and killed it again and it unlocked. So make sure you save before so you don't miss the achievement.
Locations of the 10 dragons:
Hinterlands: North of the Dusklight Campsite. Type: Fire dragon
Storm Coast: You see this Dragon by the beach but he flies away and leaves you with a Giant. After a few main mission you can get a mission from the War Table which want you to deal with the Red Lyrium Templars in the Storm Coast. This cost power to do on the war table. After you are done with that mission there is a boat which will take you to a remote island where the dragon awaits you. Type: Electric dragon
Crestwood: Go to the dam and drain the lake. You will see the dragon fly over you but you can't fight it yet. Travel to Three Trout Farm Camp and go south until you find the dragon. The dragon will sit atop of a ruin. Type: Lightning dragon
Exalted plains: The dragon is in the north part of the map. You have to go to the war table and remove the rubble to gain access to the dragon. Type: Lightning dragon
Emerald crove: North of the map. Travel to Direstone camp and keep going north. Type: Ice dragon
The Western Approach: This one you get for doing side quests for the reasercher. The quests is Draconology, How to lure a dragon, white claws, hunting patterns and a manuscript mission.
Last you will unlock The abyssal high dragon quest which make you face the dragon. Type: Fire dragon
Emprise Du Lion: In the north east there is a broken bridge. Take the keep from the red templars to the north west and you will get a mission to repair the bridge from the war table.
Cross the bridge and the dragon is on top of the colosseum look alike building. Type: Ice dragon
Emprise Du Lion: Same as above but another colosseum further away. Type: Ice dragon
Emprise Du Lion: Same as above except this time you drop down in a hole in a colosseum. Type: Fire dragon
Hissing Wastes: Travel to Sand Crags Camp and head east. You will arrive to a moutain and there is a small path between two giant stone figures. Head inside and you will find the dragon sleeping in a open area. Type: Fire dragon
Edit: You dont have to kill the 10 dragons in one playthrough, its cumulative. So if you want to you could kill the easier ones 10 times. I thought the one in Crestwood was the easiest my self. /Thanks BECKS THE NAME