Well I have two characters so far and a third planned, so I'll humor this, though it'll be pretty long:
Glaeke, the Breton Warrior-Mage, was on a pilgrimage to the College of Winterhold in order to hone her magical skills. Knowing little about the geography of Skyrim, as she stumbled across the border from Cyrodil she noticed a few Nords engaged in battle with Imperials. It didn't take long for them to mistake her for being with these so-called "Stormcloaks" and a stray fireball knocked her unconscious.
Mordsuul, an extremely, unnaturally long-lived Dunmer who was a high-ranking member of the Mages Guild during the events of Morrowind and often confused for the Nerevarine, moved to Cyrodil shortly after the aforementioned Nerevarine left for Akavir. In Cyrodil, he became the Arch-Mage of the local Arcane University. He was known throughout both provinces for being an extremely powerful sorcerer and conjuror, feared by most men. However, shortly after the events of the Oblivion Crisis, Mehrunes Dagon decided to enact revenge on the Arch-Mage because coincidentally, not only had he resembled the Nerevarine from Morrowind, he also strongly resembled the Hero of Kvatch. Mehrunes stripped the Dunmer of his magical abilities and, ashamed at his loss of power, he fled back to his homeland. However, it was around this time the Red Mountain erupted and the Argonian's began to invade, forcing him to flee to Solsteim. After spending many years in solitude on the frigid isle, Mordsuul decided that he would attempt to regain his former power by visiting the College of Winterhold in Skyrim, the last remaining bastion for mages in the nothern provinces. On his way across the border, an Imperial caravan ambushed him, accusing him, a Dunmer of all people, of being a Stormcloak sympathizer. He didn't even know who these Stormcloaks were, let alone knew why he might be a sympathizer, but the Imperials wouldn't listen and, being mostly powerless, he was knocked unconscious and wrangled into the back of a cart with Ulfric.
And last but not least, the character I have yet to make (props go to a post I saw the other day made in this style, I can't remember the user's name though):
Ha'dar travels with M'aiq. He does not know where M'aiq plans to go, but he follows. It gets colder and colder, and Ha'dar is glad for his coat. Suddenly, M'aiq tells Ha'dar to stop. Nords, he says he sees. M'aiq tells Ha'dar that these Nords are wanted by Imperials. Ha'dar has no love for Imperials, but he knows these Nords would treat him no better. Ha'dar figures he could sneak up to them in the middle of the night and capture the leader, grab the favor of the local Imperials. Little did he know, Imperials were planning to attack all along! Ha'dar was about to stick his dagger in the back of one most unruly Nord, something like Ulroc, or Ulfberth, Ha'dar did not catch his name, when suddenly Imperials storm the camp! Ha'dar did not know what to do! He looked around for M'aiq, but M'aiq was gone! M'aiq always said he preferred to adventure alone, Ha'dar never believed him. Imperials thought Ha'dar was allied with Nords, and before he could explain the situation Ha'dar was bound and blindfolded to be taken away to some execution! Horrible first day in Skyrim. Ha'dar wishes he never left Elsweyr.