Well, I might as well repost my older post about Conner's background.
Spoiler
Conner Lone-Wolf was born to be a hero.
The youngest son of Griever Lone-Wolf and Victoria Thorne (better known as the Nerevarine and the Champion of Cyrodiil) and the younger brother of Kyra and Silmarrion, Conner KNEW he was destined to become a legend. Unfortunately for him, his father was more than content to keep him and his siblings confined to Bruma, where he was Count, mostly to protect them from the Thalmor and their agents.
Most of Conner's childhood was spent in Bruma, wandering the streets, looking for anything to do. Rarely, he would get a visit from his mother and sister, but for the most part he was confined to the castle. He was a very lonely child, with no other children his age in Bruma, save for his brother and occasionally his sister. Instead, he found friendship in his father's old allies (Veran, Stalks-In-Shades, Cimmeria and Talin) whom he thought of as Aunts and Uncles.
Part of what fueled his dreams of adventure were the stories of his father and mother's exploits. Night after night he was regaled with tales from their younger years (mostly told by Stalks) despite Griever's insistence, and day after day his wanderlust grew. His father one day finally acquiesced and began training Conner and Sil in the various skills they would need to survive as adventurers. Much to Conner's dismay, Silmarrion appeared to be much more skilled than he was, as Conner only seemed to lag behind his brother in every subject, save archery. However, soon after they began the lessons ended, as Griever began to have horrifying nightmares that stole his will to sleep.
Eventually the nightmares became too much to deal with, and Griever left Bruma to travel to his home town, Dawnstar, having a suspicion that it was the source of the nightmares. After month passed with no word from their father, Conner and Silmarrion decided to travel to Skyrim to search for him, despite Veran and Talin's disapproval.
So, the young man, who dreams of being a hero, born into a legacy of heroes, heads to the north, to Skyrim, ignorant of the role he is to play in that nation's history.
TL/DR: Conner the son of my two former Heroes. His father went to Skyrim to stop Dawnstar's nightmares, but went missing, so Conner went to find him, but then DRAGONS!
As for why he fights like he does (Sword + Shield, Bow), mainly it's due to necessity. Like in the above spoiler, the only real skill he improved was his archery, which of course comes in handy fighting things that fly. The sword and shield came later, again by necessity, during the fight with Mirmulnir, when he picked them up from a fallen guard after running out of arrows.