Of the many, many characters I've created, the ones I've stuck with longest and keep going back to:
Breton female. Nature-loving wildcat. She's friends with bears, wolves, and saber cats (mod makes this easier). Doesn't get along well with most people, though. She lives in the woods, going into cities only when she must. She dual wields daggers, is decent with a bow, but has no patience for sneaking. Light armor and alchemy are her other skills. No magic otherwise. Dual daggers is even more fun than I expected it to be, and once I found the patience to really get into Alchemy, it was very rewarding.
Nord male. No magic, not even to use enchanted items. No potions, only food and rest to heal. He accepted the blessings of the divines and his Dragonborn abilities. Light armor, smithing, sword and shield. No other skills. This was a fun challenge. Not using magic at all made shouts much more important. Really recommend trying a character like this.
Nord female. Frost Mage. Since (more than?) half the foes in Skyrim are resistant to frost damage, this is also a great challenge. No armor but the flesh spells and mage armor. Destruction, alteration, and a little restoration are her skills, with some enchanting. (Enchanting gets so powerful so quickly, though, so I try not to use it much.) She has a dagger, but only uses it if she really has to.
These builds aren't very original or that creative, I know, but they are the ones I've had the most fun with. The more creative ones were too weird, got too powerful, or just didn't click, even though the characters were interesting. Or, most often, the game mechanics didn't really allow what I was trying to do. I lost interest quickly in stealth archer, conjuration mage, restoration and heavy armor paladin-type builds. Not enough excitement with those, I guess?
Even though those characters above are human, I still really enjoy playing Khajiit and Argonians for their stories. They come with conflict and a different perspective, being such small minorities in Skyrim. And you get to hear the rudest insults from people trying to kill you! The lore of those two races is fascinating. A lot to draw from there when creating a personality.