Achievements such as the Monk's "Like Water" (Dodge 15 attacks in a row) don't rely on a class-specific skill. Does this mean that they can be earned with any class?
You must be playing the class in question to receive their achievements, even if they have general triggers. (Another example is the Demon Hunter's Longshot, requiring you to kill 100 monsters from over 45 yards away - if you're not a Demon Hunter, you can't get it, even if you're a Wizard routinely killing things from offscreen.)
There definitely are achievements you can get even playing different class.
My experience: I playing Wizard, my friend playing a Barbarian, we were killing Belial if I remember correctly, he finished him with his throw weapon skill and got and achievement for that, but more importantely so did I.
My hypothesis (speculative) it only works with those "finish a boss with something" achievement (I think there are 1 or 2 more in the game)