Talk to Galmar Stone-Fist
Turn once more to the Stormcloak Camp. Talk with Galmar Stone-Fist
Get your Orders from Ulfric Stormcloak
This is a massive battle and it will take some time for you to get through. Getting through the first part of the town is not too bad. There are only a few soldiers in this portion. Plus, you have Ulfric Stormcloak and the whole of the assault force nearby you for this part. You need to move slowly and thoroughly against your opponents here.
Do not work toward the lower portion of Solitude, but push upwards. This will lead toward more to the Palace itself.
Break down the barricade set up at the archway. Fight on through the soldiers in this courtyard and make to your way to the West. You will find the entrance to the Castle Dour. This will press you forward in the quest.
Force General Tullius to surrender
Inside the building you will find Legate Rikke. She will try to dissuade Ulfric Stormcloak. Now, just wait for the conversation to finish. AFter that, attack both the Legate and the General in turn. Kill Rikke and Tillius will call for a stop.
Execute General Tullius
After you have defeated Rikke, Tillius will call for a stop. He, Galmar and Ulfric will talk. Tillius will hint that the Thalmor are responsible for the Civil War. You are now given a choice. The dragonborn and execute Tullius, or Ulfric can.
Ulfric will then ask you to accompany him for a speech he has to present. You can either tell him you will accompany him or to just leave you out of his speech. Make your choice.
If you decide to follow him for the speech, just stay with him. After just a few minutes, the speech is over. The quest has been completed and the Civil War is over.