That is the main way I have played Skyrim for years now actually. I come up with a backstory "headcanon" for my character and then play accordingly. This extends to many things, some specific to each run, but most notably the fact I do not fast-travel at all, and try using the map as little as possible.
Every time I stop by a town or city, I first go to the inn, sit at the bar, buy a drink, rent a room and actually go to sleep. Then, the next day, I go around meeting the locals, stop by the market, visit the shops.
I also make a point to try not "metagaming". For instance, one time I played a character focused on magicka, and even though it was always my plan to go to the Winterhold College, I only actually made my way there once an NPC mentioned it to me.
This has made my experience with the game like. One hundred times more engaging. I still have backups of some saves, that I will periodically open up to reminisce on. I am not exaggerating when I say I have felt emotional at points. Killing dragons and leveling up is fun, but sitting in the house I built, surrounded by the trophies of my adventures, next to my werewolf wife, watching my two adoptive daughters play? That hits different.
-- The Butch