Make your way to Whiterun and up to the Palace. Go inside and then talk with the Jarl. You will need to deliver the message and the missive with the information that is only for the Jarl.
He will consult with his advisors and then send the Dragonborn to Windhelm with an Axe. Head for the Palace of Kings to immediately arrive in the location you need. Go inside the Palace and find Ulfric Stormcloak. Talk with him and give him the axe.
Deliver Axe to Jarl Ulfric
With the Axe in hand, it is time to head to Windhelm.
Return to Whiterun and warn the Jarl
With the message delivered, make your way back to Dragonsreach. Talk with the Jarl once more. This will begin the next mission.
Report to Legate Quentin Cipius
Having talked with the Jarl, head up the stairs behind the throne. You will see a strategy table there with the person you need to speak to Legate Quentin Cipius. There will be a messenger who arrives about the same time. They will report that the Stormcloaks are massing outside the city to attack it.