After defeating Clair, enter the Dragon's Den-just north of the gym, over the water- then surf down and left, around the outside of the cave, until you come to a small step leading into a house floating in the water. Enter the house and defeat the trainers.
You will then be asked by Clair's grandfather questions testing your respect of pokemon. If you answer them all "compasionatly" then he will give you that Dratini with Extremespeed.
(You may need to talk to him twice, as you are interupted by Clair and possibly Lance as well).
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You just gotta act like Ash (with his "I love Pokemon!" Attitude).
Only in Pokemon HG & SS is one of dratini's egg move and you can get one if you complete the elder's quiz in dragon's den in blackthorn city in the Johto region
Dragonite doesn't learn extreme speed through level up, nor from other pokemon by egg breeding. The only way to get a Dragonite with extreme speed is from HG/SS version in Dragon's Den, after giving the old geezer guy inside the right answers for his test. If you answer correctly he'll give you a Level 15 Dratini with extreme speed. However, giving him a slightly wrong answer he'll give you a regular Dratini.
you cant