I've not given it any real thought, but off the top of my head, I guess she's the sister of the Oblivion hero...assuming Bosmer live that long. If not, then she's his daughter or niece or some other thing. As for why she went...maybe she got sick of living under his shadow all the time, like that Giselle or whatever who's the sister of that guy in the palace in Solitude. Or whatever it was.
So she moved to Skyrim with the intent of expanding the family potion business. I don't know why she was being executed...stole a sweetroll, probably.