New Main Objective: Talk to Galmar
Continue talking to Galmar, asking him about the details of the next assignment (the above screen). He tells you that a unit under his command is preparing to set off for a place that supposedly holds the titular Jagged Crown. The Stormcloaks wants to claim the crown before it falls into the Imperial Legions' hands. Galmar suggests that you meet him near the ruins of Korvanjund.
New Main Objective: Meet Galmar outside Korvanjund
Leave the Palace of the Kings and open the world map. Your destination isn't Korvanjund directly, but a location some distance away (the above screen). I don't recommend using Fast Travel (if you unlocked that location earlier), but travel there on foot. The ruins lie in the central part of the map and you can get there either from Windhelm (go west) or Whiterun (go northeast).
Once you're there, look around for Galmar's unit (the above screen) and join it. If you worked with Rolaf before, you'll meet him here. Talking to him doesn't change anything, though. Wait until Galmar approaches you, and in the meantime you'll be told that the Imperials are already here. Confirm your readiness to attack them.
New Main Objective: Retrieve the Jagged Crown
Follow the other Stormcloak soldiers until you reach the Legionnaires camp. If your character prefers close combat, run into the fight and get rid of enemy archers first (the above screen).
If your character prefers ranged combat, don't use the stairs at all but simply eliminate enemy soldiers from a distance (the above screen).
No matter the tactic, remember not to use area-effect weapons, abilities or spells, because there's a high chance you can hit your allies, and they may strike you back. After the battle, follow Galmar and his men inside the ruins. Listen to a short speech (the above screen) and open the door to enter Korvanjund Temple.
Crouching, join the other Stormcloak soldiers (the above screen), but attack the Legionnaires only when Galmar gives the command. When you fight inside the ruins, be careful not to accidentally hit other Stormcloack Rebellion members. Don't worry that they would fail without you, because Galmar's units should be able to win battles even if your involvement isn't crucial.
After the battle, wait until Galmar posts two soldiers there and follow him. The next battle starts when you reach a larger room (the above screen); use the fact that some foes will have to use the stairs to join the main fight.
Fight until all Imperials are eliminated and go downstairs. Galmar should stop at the north entrance (the above screen), saying that there is likely a trap in the next room. He suggests finding an alternative path.
If you feel secure and so far hadn't had any problems with eliminating enemy soldiers, you can ignore Galmar's warning and go north right away. His assumptions about the trap prove to be correct (the above screen) and you'll probably have the most trouble with the archers, which are not easy to reach. Luckily, your allies join you in a moment, so you don't have worry about fighting them all by yourself.
It's much better to follow Galmar's counsel and look for an alternative path. Return to the stairs and once upstairs, go east (the above screen). This way, you will go around the whole place where the previous battle has been fought, and reach a corridor that leads to the adjoining chamber.
Crouching, go into the north chamber. You should see that two or more archers are occupying the balcony to the southwest, so it's best to get rid of them by aiming at the lamps above them. Once they fall to the ground, they cause a fire right where the archers are standing (the above screen). Depending on your playing style, stay on the upper balcony or jump down and help the Stormcloaks eliminate all enemy units.