Personally, I RP my characters as much as I can get my mind into it.
They have an eating schedule, sleep during certain times (unless I'm a Vampire {which most of my characters are...}, then I RP that Vamps don't need sleep...), since they are mostly Assassins I'll have them pray to Sithis once every three days after they've killed an innocent, they change into "normal" Fine Clothes before entering a City/town/Village so that no one sees the Dark Brotherhood Shrouded Armor, etc.
To continue this:
With my Assassin's (most played character archetype), the one I just started last night has a full stock of food including bread, a cooked meat (usually venison or fish as they are relatively easy to acquire whilst in the wilderness), Eidar cheese (my characters' personal favorite), Honningbrew and Black Briar Mead, a tankard, and usually a Sweetroll for desert. He only eats once per day (an evening meal), and I'll wait an hour after eating everything to simulate the time taken to enjoy a hot supper.
He sleeps typically during the day and handles his shopping trips during the late afternoon hours.
I don't fast travel with him, though I do take the carts every so often.
He carries two sets of Fine Clothes just so he can change into a clean set every so often.
He uses daggers and bows only, and genuinely spends a majority of his time in the Sanctuary when not out handling business.
Clarifying the topic above of his devotion to Sithis, my Assassin (Nocturne, BTW...

) doesn't just pay homage to Sithis after slaying a random person, he actually goes out and finds a victim and uses their murder as a sacrifice to the Dread Lord, and takes everything from their corpse. He leaves a solitary sprig of Nightshade behind in the corpse's inventory to designate that Soul to the Dread Father.
He has a cover business as a traveling Blacksmith, and so he works in the various cities that he travels to as a way to keep up a visible income that way no one would guess his real "job".
Because of the destruction of his homeland, he has a severe bias against Argonians. How got over his wrath towards them long ago, but he still hates them and kills them as he sees them. He slowly stalks them, plans the perfect time, and leaves them as his sacrifice to Sithis. He views this as helping to rebuild Morrowind by removing Morrowind's enemies, one at a time.
He takes no Followers as no one who sees him in his Shrouded Armor can be allowed to live. Also because of this rule, he currently has a body count of 35 City Guards... but he's stealthy so he has yet to acquire a bounty.
And that's basically how I play my Dunmer Assassin, Nocturne.