Actually just make it like in Dragon Age:Origins.
Give every race it'S own background story and add a few remarks to this background every now and then in the mainplot.
I don't see any big difficulties here.
Human? Well you start at a military Academy and go on a training mission as part of a x person squad.
Depending on class you take another spot in the team e.g. biotic, soldier or engineer and somehow something goes wrong and you end up in the main storyline.
You could continue this for every race
Batarian? You are part of a slaver crew and try to capture a spaceship but all goes wrong and you end up in the mainstoryline.
I would also add the option for everyone to refuse to take part in the main storyline ... lets say "amnesty for whatever went wrong in the prologue or life sentence in prison" if you choose the later let it be a game over and create an autosave before the decision.
Once in the storyline the person goes back to his homeplanet or whatever and faces some minor consequences .. a Krogan gets challenged to a duel or stuff like that.
If i was working for Bioware i would let the community send in special background stories and chosse whatever i think fits best.