I just started a new character today, they've just joined the Stormcloaks, and so far I've only just begun their first quest.

Anyone have any ideas of fun roleplay I could do once I've run out of their quests? Any side-quests that involve them I wouldn't normally be able to find? Any roleplay objectives? (Things like raiding imperial camps around, etc)

Other than fighting Legion soldiers (like you said), I can't see what else you could do.

Kill all Imperial soldiers, Thalmor, and Nazeem.

How could I forget about the Thalmor?

Don't many of them disappear? I suppose you could frequently visit the Embassy.

http://www.uesp.net/wiki/Skyrim:Missing_In_Action

And even though the whole quest is not exactly linked to the Stormcloaks, you can complete http://www.uesp.net/wiki/Skyrim:In_My_Time_Of_Need by aiding Kematu and RPing that you are making friends in Hammerfell.

Unfortunately there aren't many exclusive quests. Really only the recruitment quest, since the two sides of the war pretty much parallel each other. There will be plenty of imperial camps, both before and after the war.

There's a quest called Missing in Action that's related, though as I said not exclusive to Stormcloaks. You get it from Fralia Grey-Mane in Whiterun.

The Dark Brotherhood quest line intersects in an interesting way at one point. So does the main quest. If you've never seen Season Unending as a Stormcloak, wait until after the Battle of Whiterun to pursue the main quest. There's some unique dialogue, Vignar is there and Tullius is the one who starts off with big demands.

Use a weapon with soul trap so those imperial loving nords never see Sovngarde.

Drink mead

Lots

Of

Mead

That and I always enjoy slaughtering those patrolling Thalmor with the Nord prisoner.

kill every imperial patrol you see. oh and thalmor patrols. don't worry about the bounty if there is one. last witness killed removes that bounty. Clear all imperial camps. If your on the PC use the mod Kill Them Generals. it allows you to kill the imperial commanders at each camp.