Before we get started, I just want to clarify that this walkthrough will be rushing through all of the answers to the keypads in the game as this is the only requirement for the "Solve All Puzzles" achievement. I will not be explaining where I got any solutions, nor will I be explaining any other puzzles. This is solely an achievement walkthrough.
Using this walkthrough, you can complete the game in under 15 minutes of gameplay easily. The only thing that increases the time is that you must wait 70 minutes for a timer to run out before finishing a run for 1 achievement.
If you lose one of the keys somehow, you can pause and press Reset Items for it to reappear where it initially spawned.
Now, let's get started!
As soon as you get into the little starting hub area, look left and use the screen to select Solo and Start a new game.
When you spawn, go around the room clockwise entering the following codes in each keypad you find:
8390
4295
6915
4958
8998
1082
Grab the key sitting at the edge of the bed, then use it to unlock the door that doesn't have a keypad. Click it again to go through it.
Open and walk through the door in front of you.
Pick up the blood bag on the bottom shelf. Turn to your left, and click on the only open space in the 2nd row of blood bags. It will place the blood bag you picked up in this slot.
This should unlock:
MeticulousNever Assume, there is a reason for everything
Use the following code on the only keypad in this room:
5649
Turn right and click on the fuse box to the left of the door to open it. The top row of fuses should be set in the following order:
Up, down, down, up, down, up, down
The lights should turn on. Go out into this hallway and put the following code into the keypad on the right door:
7413
Go through the door.
Start going through the room clockwise putting the following codes into each keypad:
9362
1981
NHSW (2609)
Open the final cabinet (that says 2609 on the keypad) and pick up the key inside (try pressing 
and trying again if it won’t pick up). Go back to the door and use it to leave into the hallway.
Use the key on the door to your immediate right, then enter it.
Again, travel around the room clockwise entering each code in the corresponding keypads (and single directional pad):
Left, up, up, right, center
135
3922
Leave the room and head back to where you spawned. Go through the door that has the code 6915 on it.
One final time, go around this room clockwise and enter the following codes in the keypads:
1793
7290
3527
Open the 3527 cabinet and pick up the key inside.
If you pause, you'll see a timer counting down on the top right of your screen. You need to complete the game at least once after this has run out to unlock an achievement. I am going to assume you did this on this run. If you do not wait it out, make sure to do this on another run later!
Use the key you just picked up on the other door that's in this room (to the left of the 1793 keypad) to finish the game.
This will unlock:
Now that you are back in the hub. Click Multiplayer and then the Create button. Start a new game.
Once you spawn, turn right and enter 6915 into the door keypad. Enter the door.
Open the low cabinet immediately to your right by entering 3527 into the keypad. Pick up the key and use it on the other door to end the game again.
If you do this before that timer from the pause menu hit halfway (and you should have because it's like 35 minutes), this will unlock all of the following:
I am SpeedEscape the office in less than half the time
The only remaining achievements you should have left are to finish the game in multiplayer. This will take two more playthroughs.
Once you create a multiplayer lobby, invite 1 other player. Complete the game real quick using the Multiplayer Solo playthrough that I detailed above.
This will unlock:
Hold my handEscape the office with exactly 2 people in the party
Repeat the steps above one final time with a total of 3 players in the lobby to unlock:
This should officially unlock the final achievement!