For this achievement you must defeat all 5 Dragons scattered around the world map, after defeating all 5 you will receive the achievement, this is also necessary to obtain the Dragon Bracelet achievement.
1. Moody Dragon: Found in the Underground River, where you will see a TON of Kelolons, you must defeat ALL of them, once you do that the Dragon will appear.
Battle: Very easy fight simply just lay into him with high powered physical attacks and he should go down without a hitch.
Note: If you haven't done so already you will need to visit Kelaso Village and complete the side quest there before the Spark Dragon will appear.
2. Spark Dragon: Found in the Gul Mountains, take these directions from the Warp Device: left, right, right, left, left.
Battle: Obviously weak to water, keep up the high level water spells and Corporeal attacks to take him down. His attack is pretty high so make sure you are well healed as well.
3. Phantom Dragon: Found in the Great Desert on the southern-most point. Warp to Lal Mountain Camp then fly there.
Note: If you are going for the Complete Item Record, you MUST steal Nene's Earing from him, you only get one chance to steal at the very beginning of the fight and if you don't get it then reload your save and fight him again until you steal it on the first try.
Battle: This dragon is ONLY weak to Wind attacks anything else won't work or will heal him. Use Windus or Windus sword to take him down and stay fully healed. Also do not use Corporeal attack as they will heal him as well.
4. Blizzard Dragon : On a small hill west of Kelaso Village.
Battle: Attack him with Flarus and Flarus Sword only and keep your allies healed.
5. Amethyst Dragon: Exit Jibral and head South-West, follow the road to the coast, then follow the Coastal Road around to the dead end and the Dragon will appear.
Battle: Not necessarily weak to anything, just keep up the physical attack. Magic Sword works great as well as Charge Attack. Corporeal attacks also work great here as well.
Being between Level 70-80 is more than enough to take these guys down without much of a problem, anything lower (besides the Moody Dragon) will prove to be a tough battle.