This Achievement is more or less inside of the Whiterun storyline. As you complete quests for Jarl Bulgruuf the Greater (Or Jarl Vignar Gray-Mane if the Stormcloaks have taken over), you are slowly moving to this epic climax of questing.
Now while this Dragon is a boss, there are many ways to defeat it easily. I found that the best way is to just move around the tower getting cover from his fire breath. All the while shooting a steady stream of arrows towards it. During this, the Whiterun guards and Irileth might actually assist you.
All in all this is a very easy 10 Gamerscore, and may be-if you're like me- one of the first achievements you will earn from The Elder Scrolls V: Skyrim.