Dragon Rising

Talk to Jarl Balgruuf
Talk to the Jarl of Whiterun about your reward for the previous quest and you will be interrupted by a guard. He says a dragon has been sighted by the watchtower near Whiterun. You and the Jarl's housecarl Irileth are told to take care of it.

Meet Irileth near the Western Watchtower
Walk out of the city and meet Irileth there. She will tell you to follow her to the tower where the dragon has been spotted. The tower is not too far away from Whiterun. Once you arrive you will find the tower has been mostly destroyed by the dragon. When you walk up towards the tower the guards around it will tell you that the dragon has left but will be returning for another attack.

Kill the Dragon
Once the dragon returns you will have a difficult fight on your hands. It can be very difficult to attack it at first since it will mainly be flying around predominately. Wait until it lands to attack it and try and use power attacks or destruction magic to cause a good amount of damage because it does not stay grounded for long. Also, try and avoid its fire breath while attacking or while it is flying in the air as it could cause a lot of damage to you, and unless you are strong in Restoration you could die very easily from it.

Investigate the Dragon
Once you have defeated the dragon be sure to loot it. Dragon Bones can sell for a lot of money (and are a rare smithing ingredient), but be careful how much you take because they weigh a significant amount and can weigh you down easily. You will also notice that once you are done looting you will absorb the dragon's soul. Once absorbed the guards around you will begin to call you Dragonborn. You now have the ability to shout and use that ability.

Report back to Jarl Balgruuf
Irileth will tell you to return to Jarl with the news that the dragon is dead, and that you may possibly be the Dragonborn. While you are returning back to the city you will hear a number of shouts coming from the mountain.

The Jarl will tell you that those are the shouts of the Greybeards and that you must travel to the summit to see them. The Jarl will also reward you with the title of Thane of Whiterun and give you a personal bodyguard and you may buy a house in Whiterun for 5000 now.

Der Aufstieg des Drachen