Me? Well usually, I think of a basic idea, such as a Spellsword, or thief. Then, what would they do? Who would they fight for, if they would even fight? Then I get them a name, usually using the Tamriel Rebuilt name generator.
I'll describe my latest character's thought process, to give a better idea:
since playing again recently I have done a few basic types, Warrior, mage, courier/fisherman. So I figured, why not a thief? Then I thought, "hmm, an Argonian thief!" So, then I created the character. Before entering the portal to Cyrodil (Using an Alternate Start mod), I thought, will he be in the guild or not? So I decided he'd be independent, and having added an extra fence in through a mod so I could pull off the independent act, I begun to play. Some details came along as I played, such as, he likes to dress sharply, and simple 2 gold pants didn't cut it. He had expensive tastes. He died to the imperial watch before I could figure out much more than that, though.
So, how do you do it? Do you make a character as you start up a new game? Spend time planning every little thing out? Or a mix?