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You'll receive this quest immediately after completing the Stormcloak version of the quest
Message to Whiterun
After learning that Jarl Balgruuf of Whiterun has thrown in with the Imperial Legion, Ulfric will send you and his other soldiers to Whiterun to capture it. At this point the map nodes for the Whiterun Stables, Whiterun and Dragonsreach will disappear, and you'll have to travel to the Whiterun Military Camp, which is located just to the south of Whiterun.
When you arrive at the camp, you'll find Galmar and some Stormcloak soldiers (including Ralof) waiting for you. Galmar will give a speech to rally the troops, and then everybody will charge at Whiterun. Don't worry too much about fighting the Imperials during this sequence. Dead soldiers (on both sides) will just get replaced, and they won't leave behind bodies that you can loot (unless you're really quick). Instead, break through the initial barricades (#1) and then follow the path (#2) to the top of the drawbridge (#3), where you'll find two levers. Pulling either of the levers will open the drawbridge and force Whiterun's defenders to retreat (via #4) into the main part of the city.
Inside Whiterun, you'll have to do some more fighting, but enemies won't re-spawn, and they'll leave behind corpses when they die. When you get to Dragonsreach (#5), you'll encounter Jarl Balgruuf, and you'll need to fight him until he surrenders and orders his men to stand down. This will end the battle and cause Balgruuf to leave the city. Vignar Gray-Mane will then become the new jarl, and Whiterun will return to normal -- that is, the NPCs will come back -- once you've informed Ulfric of your success.
1 - Barricades
2 - Path to Drawbridge Levers
3 - Drawbridge
4 - Inner Gate
5 - Dragonsreach