This is NOT an easy achievement in any way.
First of all you will have to unlock the speed run mode by beating the game. Beating the game requires you to find all puzzle pieces and solve all puzzles in order for you to unlock the final level (or world 1).
I recommend if you have not recently beat the game to play through the whole game again without a guide to get yourself familiar with your own way to solve the puzzles before looking to see if there are faster ways of doing them.
If you just got the game I recommend going for this achievement soon after beating it so you will be familiar with a way to solve the puzzles.
When you are going for this achievement plan to spend some time on it, I know the achievement says you have to do it in less than 45 minutes, but you have to become very good at solving each of the puzzles in order to do it that quickly. Personally I spent 7 hours retrying the full game speed run before I finally got the achievement at only 43 minutes.
EDIT: The final stage of the game will take a MINIMUM of 4 minutes, that's without messing up at all so if you're going through the final stage at 42 minutes or more you can go ahead and restart, chances are you will end up with just above 45 minutes. Also if you get that close you should look at the video and review any stage you had difficulty with.
If you are looking for ways to do levels quicker I recommend looking at this guys speed run. He did it in 32 minutes and NEVER messed up. I used some of his ideas and saved just enough time to get the achievement.
EDIT: Don't worry about memorizing the whole speed run, you CAN pause the game without the time adding up, so if you are having trouble with a specific puzzle go ahead and pause the game and then look at the video until you know how to do it, also putting the puzzles together does not count against your time so don't get anxious or nervous solving the puzzles.
There are 3 parts to this video, you can find the other parts through the same video.
EDIT: Decided to put all three on here as the comment suggested