Your question is hard to clearly answer because it seems to be based in false assumptions and broken understanding. Once you complete a part of the story, at any time, you can go back and hop around to get the various achievements. The problem is: What do you define as a 'playthrough'? If you are looking to estimate the time it would take, due to the random nature of dungeon and event generation and positioning, this would be a flawed metric.
If you consider a base definition of a playthrough as playing from the start of act 1 to the end of act 4 sequentially, not accounting for re-instancing the areas to farm dungeons, drops and events, the absolute minimum would be 4 hardcore multiplayer runs for each of the 10 characters on each difficulty, assuming everything worked out perfectly and you found all the random stuff and all your characters leveled to 60, making for a total of 40 playthroughs.