The playable character is actually composed of two different characters, a mortal and a Daedra, sharing the Daedra's body. I will here expand their backstories here and expose various propositions for the mortal character.
For Altmer and Bosmer he calls himself "Calemryon", for Dunmer he calls himself "Calem" and to his friends he is just "Cal".
Calemryon is a brilliant mage of mixed Elven origins, and lived in Summerset Isles. When the Thalmor went to purge the society from those who weren't of Aldmeri blood they came for him as well because of the bastard blood (and moral issues, troublemaker…). Calemryon killed his aggressor and tried to flee but got wounded in the spine when the Daedric artifact he tried to use was shattered. A Daedric shard is still stuck in his back and removing it could mean death. The mage was never able to walk ever since. It was only with the aid of conjured Daedra to hold his body that the he could run away from the Thalmor and go into hiding.