Now THIS is an entertaining mission. Report to Galmar at the Whiterun Military Camp, where the war against Whiterun is already well underway via. catapult. There will be half a dozen soldiers here that Galmar is once again giving a speech to. At the end of the speech, your objective will change to "Break through the Enemy Barricade".
Head down the road with the stampede of Stormcloak soldiers as you reach the outskirts of Whiterun. There will be a barricade here that you'll have to smash while archers try to gun you down (simply attack the barricade to hurt and/or destroy it). Either attack the barricade yourself or let your fellow soldiers do it while you pick off the Imperials and Whiterun Guards. After destroying the barricade you can proceed, but the drawbridge to the main doors of Whiterun is up, changing your objective to "Open the Drawbridge".
To do so, use the rubble on the right after smashing open that first barricade to bet on the ramparts. There will undoubtedly be SEVERAL Imperials and guards here, and for the moment besides getting some archery support you are on your own. You can either kill all the enemies (if you're badass enough) or make a beeline for the objective and open the drawbridge. This will let the Stormcloaks enter the city proper, with the new objective of "force Jarl Balgruuf the Greater to surrender".
Enter the city and start pillaging and looti... I mean, make your way towards Dragonsreach. There will be several more barricades in place here, placed at strategic locations and pretty lightly guarded, so either go on a rampage or go with your army to fight your way up to Dragonsreach. Enter the main hold once you get up there. Here there will be five guards, the Jarl, and his Dark Elf Housecarl. You'll have several soldiers to help you along with Galmar and Ralof. Let the first wave of soldiers come to you and then move up to deal with the Jarl. You'll need to "kill" him to stop the attack on the city. I put kill in quotations since he doesn't actually die.
Once you have him beat, he will tell everyone to stand down and will talk with Galmar and Vignar Gray-Mane. Vignar will become the new Jarl of Whiterun, and will be a loyal ally to the Stormcloaks. Galmar will tell you to go and tell Ulfric of our victory. This ends this entertaining mission and starts our next one: "Liberation of Skyrim".