What I have tried to stress is that for someone new to TES games like myself. Playing Skyrim would lead you to think that it has been at some point in the past conquered militarily by the empire. The empire is presented in the game as a force of occupation just consider a few points.
One The empire is obviously at least loosely based on Rome.
Two the natives of Skyrim are based on the more barbarian tribes that Rome put a lot of effort in to attempting to conquer.
Three It comes with a map that says the province of Skyrim.
Four The natives are staging a war to push out the empire and establish a free independent Skyrim.
With only those things to go on you have to say that the civil war is not a civil war but a native uprising against its oppressive conquers. Having read what people have posted its clear that its not quite that simple.
Culturally there are similarities between real world history and The Elder Scrolls universe, but those shouldn't be taken as an indication of how things are in Tamriel, it's definitely complicated and hard to follow. The short answer is that Tiber Septim was a Nord, (or a Breton, or an imperial depending on your source, complicated) who conquered all of Tamriel and founded the Septim Dynasty later ascending to join the Divines as Talos. So Stormcloaks (and pretty much all Nords, even Imperial sympathizes, if you ask certain people) identify with Tiber Septim. They do not identify with the Mede Dynasty because it disowned Tiber Septim's ascension and thusly their cultural connection with the Empire. The Empire isn't viewed by the Stormcloaks as an occupying force, but as a weak and ineffective ruling body because they sold out on tradition to save their own skin. In fact I would go so far as to suggest that Skyrim is likely one of the only provinces that did
not view the Empire as an occupying force, the elves certainly swing that way but not the Nords.
In any case it's not simply a war for autonomy either. I highly doubt that Ulfric would stop at securing Skyrim's independence if he won the conflict anyway, someone who strolls into the King's palace and literally shouts the king's head off doesn't just go home at the end of the war and start a family. He'd likely set his sights on building his own empire next.