I finished four and a half play throughs. I didn't even get all the Achievements. It can be done, but - here's the hint - plan, plan, plan.
Don't play the "multiclass" characters if you're after Achievements, stick to Solider, Adept or Engineer.
Know what crew members you want to get the Ally Achievements for and get those allies in your party before you start on any of the other quests, including the little Citadel ones.
Enemy snipers in Insanity mode suck ass.
Seanchai