It sounds like you're asking about the first dragon you fight, outside of Whiterun, as part of the quest Dragon Rising : you follow Irileth outside Whiterun and upon completing the quest:
The Jarl makes you Thane.
If so, it's supposed to be an introductory dragonslaying encounter: if you followed Irileth around town before heading to the watchtower, you should have a fairly decent posse of guards to help you out.
To be honest, I was still learning how to use the controls on the PlayStation 3 version of the game at this point, and I essentially ran around the watchtower like a chicken with its head cut off.
The guards and Irileth will do most of the work.
I was a Conjuror build, so I summoned a Flame Atronach for good measure.
After a few minutes, the dragon will die pretty easily, whether you made a significant contribution to the fight or not.
gaming.stackexchange.com/users/None/
The basic way to kill dragon is : Attack from the ground with spells and/or bows
When the dragon is grounded, use whatever you want and try to keep it grounded with the shouts you already know When Dragon HPs runs out, dragon dies.
gaming.stackexchange.com/users/7709/M'vy
Theres a small pillar at the foot of the steps, i used a bow to take shots at it while using the pillar to stay out of the dragons line of fire.
gaming.stackexchange.com/users/15565/Deft kitteh
When he is airborne shoot with a bow.
When on land attack with with a one-handed weapon in one hand and a healing spell in the other.
gaming.stackexchange.com/users/22853/ralphy
Just run into the tower and wait inside, the other soldiers will kill him, and if he''s dead you also can take some guard clothes..
gaming.stackexchange.com/users/30721/Mika
The mission you are talking about is basically an introduction to killing dragons in Skyrim, all you have to do is shoot arrows at it until it lands, then use a 2 handed weapon to do maximum damage while it is down and with the guards help it is pretty easy to kill, if you still find it hard just lower the difficulty, get your level up by killing bandits or you can camp inside while the guards kill it.
gaming.stackexchange.com/users/30877/fred
You can kill the dragon pretty easily.
Just use a bow and some iron arrows.
The group of guards will have the aggro of the dragon, so just fire when you get the chance.
You won't take damage and it'll take about 30 seconds to defeat it.
gaming.stackexchange.com/users/33531/max
As part of the main storyline, I'm tasked with slaying a dragon outside of Whiterun. I follow Irileth and a bunch of other NPCs outside to confront the dragon, but I'm not sure how I'm supposed to kill him.  What's the trick?
How do I kill the first dragon encountered?
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<the-elder-scrolls-5-skyrim>
gaming.stackexchange.com/questions/36661
gaming.stackexchange.com/users/14572/arik
The guards and Irileth do most of the work. The dragon will die pretty easily after a few minutes. It seems that the attacking guards will kill the dragon pretty easily.
It seems that the attacking guards will kill the dragon pretty easily.