Before and After
To obtain this achievement you must go in and out of Classic and remastered mode on every mission and them finish every mission doing so (Only have to go into both modes before finishing the mission you do not have to play in them).
I like to play in classic mode, and to obtain all the skulls you have to go into anniversary mode to get them because they aren't in the Classic mode. After finishing all missions I will unlock this achievement.
To switch the modes (which takes no loading time at all) hit the View button in the middle of your controller to the left side
Notice: Its possible that if you switch back and forth and then die you must do it again until you get a checkpoint saving what you did.
Another thing to note is if you finish all the levels and you did not unlock this achievement, check your Achievement progress in the achievement app. For every 5% missing you are missing a level (5% is in Classic, 5% is anniversary. (Each level technically counts as 10% but if you play through every level you will no matter what have a minimum of 50%)
There is no exact way to know which missions you did both ways so you will have to try and guess which one you missed and mark them off on paper or Notpad on your computer/laptop so you don't keep trying the same missions.
According to AbroadShaneSolo
"You do not have to finish the level if you have already completed the mission once. You just have to go through and switch the modes and get to the next checkpoint. As long as you don't die the percent will go up on the achievement you can check on total completion percentage of the achievement."
So you do not have to run every mission beginning to end!