My Imperial is a paladin of sorts. A devout follower of the Eight, and especially Arkay, she is on a mission to destroy the undead of the land. Skyrim is full of old barrows that are crawling with draugr, so she's bringing a little light to the darkness. The light of holy, cleansing fire. She kills bandits as well because no matter what the Stormcloaks may think Skyrim is an Imperial province and as a soldier it is her duty to keep the citizenry safe.
This is pretty much my Imperial Paladin of Stendarr's motivation too. He came to cleanse the land of evil undead, daedra, vampires or werewolves because he'd heard tales of the Volikhar, Dragur and other dark forces native to Skyrim. But he's also a Legionnaire, his family had many members in the Legion and he in the soldiery for a time before finding his calling with Stemdarr as part of the Legion (He learned the art of healing from a Legion chaplain of Stendarr, as Stendarr is patron of the Legion) and he agrees to fight for them in Skyrim as well.
My Dunmer Nightblade is traveling around looking for work as an assassin or pretty much any other job that pays. He has no qualms about doing whatever he needs to gain knowledge, power and wealth.
My Altmer Spellsword has a history of opposing his Thalmor brethren, and he had assisted the Valenwood resistance until they began actively hunting him. He has connections with the Psijics, the Valenwood Blades, and the Legion, and comes to Skyrim to fight the Thalmor Justicars covertly. He'll join the Legion as well because he believes in cosmopolitanism and unity in Tamriel.
Also, they all are discovered to be Dragonborn.
