For this you have to complete every level pretty much in a single run without letting the multiplier counter reset. (exception being the one short puzzle level (7-4)). Note, this is a cumulative achievement, so it can span multiple playthroughs.
The key to this is the back button. The back button will reset your multiplier timer to full time, even if you had less than that before you made the last move. If you need time to think, quickly move a block, think and then hit the back button. Note, if enemies in the later levels move blocks, you will not be able to back out before they start moving the block.
If you die in the level, you must restart from the beginning, not a checkpoint. If your multiplier expires, hit the back button.
You can do this at the same time as the Golden Child achievement if you are playing on Hard,
but it is far easier to do this on Normal due to the back button not being locked. I honestly suggest doing it on your second playthrough once you have a hang of all the moves, particularly the Inazuma technique, but you can do it by re-challenging the levels using your cellphone. The advantages to re-challenging on the cell-phone are
- no pillows to collect
- infinite continues (only really an issue if you are trying to do this on hard).
- can not save in between levels.
Walkthrough videos are shown on the Catherine Wikia: http://catherinethegame.wikia.com/wiki/Walkthrough:Overview