Here is the full list of all 16 Dragon Chase achievements.
Reach 10000 score
Reach 20000 score
Reach 30000 score
Reach 40000 score
Reach 50000 score
Kill an Orc
Kill 5 Orcs
Kill 10 Orcs
Kill 15 Orcs
See the Game Over Screen for the First Time
Get Killed by the Dragon for the First Time
See the Game Over Screen 5 Times
See the Game Over Screen 10 Times
Kill 5 Fighters
Kill 10 Fighters
Kill 15 Fighters