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Background

To trigger this quest, you'll first need to talk to Ulfric Stormcloak in the Palace of the Kings in Windhelm, and tell him that you'd like to join the Stormcloaks. Ulfric will remember you from Helgen at the start of the game, and he'll direct you to Galmar Stone-Fist, who should be nearby. You'll then need to talk to Galmar.


Walkthrough

Galmar won't be sure about you, and so he'll decide to test your mettle by sending you out to the Serpent Stone, where he'll expect you to kill an ice wraith. You'll find the Serpent Stone in the northeastern part of Skyrim, and you'll probably have to fight tougher things than ice wraiths while making your way there, and so the battle with the ice wraith should not be all that difficult.

When you return to Galmar after completing the task, he'll ask you to swear an oath to Ulfric and the Stormcloaks. Doing so will make you a Stormcloak. For your reward, you'll receive Fur Boots, Fur Gauntlets, a Hide Helmet, and a Stormcloak Cuirass.

Note: Completing this quest will cause you to fail the "Join the Imperial Legion" miscellaneous quest, but you'll still be able to join the Legion (and betray the Stormcloaks) later.