I think this means I have to play like 3 more times to get all the achievements, but maybe I'm wrong.
You need a bare minimum of 3 playthroughs (plus some partial games) to get all achievements, and that's if you had a plan to do so before you ever started playing the game (three different ally combos, normal/hardcore/insanity, minimum of two games to hit 60, and you need to spend at least some time on all three pure classes to get all the mastery/expert achievements).
Took me 4 completions (first one was a soldier, who got the story-based achievements, completionist, power gamer, a few of the weapon expert ones and various other easy ones like Scholar; second and third were a vanguard, and got me 4/6 ally achievements, paragon, extreme power gamer, Hardcore, and more weapon expert/biotic mastery achievements; fourth was a quick blast through on Insanity with my level 60 vanguard ignoring most of the sidequests), plus another mostly-complete game (with an infiltrator, for the final two ally achievements, renegade, snipers and some tech power achievements; i'm finishing this game off now to actually see the full renegade ending), plus a couple of partials (for the expert/mastery achievements i needed that you can only get with an adept/engineer).