Aground

Keep completing the Alchemist's quests. He is the guy in the purple robe first found underground to the right of your starting base, just before the stone mountain. The Dragon Lair is found in that stone mountain a little up and to the far right (east) of where alchemist's lair is.
The quickest way to get this going is to get a shield and sword and a net. So you will have to have unlocked the merchants (which is the pre-cursor to the boat doc) so if you aren't that far yet just keep doing all the quests to advance the game. Once you beat a wyrm down to low health like 10%-20% left, place a net in front of the wyrms path. It might be running away from you at this point so just run in front of him and place it, or if it is still chasing you place it immediately and back up. If the wyrm is low enough health this will capture it. (FYI you are going to need 5 for the quest and you'll want at least 2 more to keep an automated supply of dragon blood going.) But for now take that 1 wyrm to the Alchemist and he will give you a quest to find the dragon lair. Once you go to the dragon lair find the spot in the upper left of that lair where you can summon the alchemist, and he will make it possible to talk w/ the dragon. The Dragon will give you a quest to bring him 5 baby dragons. You will need to feed 5 baby wyrms 8 steak or coal to get them to change into baby dragons. (which you can do in advance to make this quicker), once you talk to the dragon again he'll take the baby dragons and you will have control of the lair. Click to manage dragons and when the menu pops up select fly dragon. Your character will be mounted on the dragon and you just press up on the stick to start flying and pop goes the achievment.