This guide assumes you have beaten Kaptar first, and is full of spoilers!
Kaptar is the place with all the windy mountains and chains!
If you are not done with Kaptar, you can not follow this guide!

First, head to this room. You'll find a piece of paper with a 20-digit code.

This code is different for every game, so write it down

If you're having trouble finding this room, it's close to the hive.

Go back to the machine with all of the Russian text on it, near the wind turbine.
Make sure it's connected to the stairs, like so

If it's not connected, you have to raise the stairs again using the crank at the top.
Lower the stairs while the device is still connected, to lift the machine up like this:

Now, enter the 20 digit code from before. It's a little hard to stand at the right spot for the buttons to line up with the cursor. When you're done, hit Enter (the very bottom button on that row.)
It doesn't matter which lights are turned on, all that matters is the numbers displayed at the top.

You should have just earned the "Sweet Revenge" achievement, and the machine should be nice and busted!

Now that the machine is broken, enter the following code and hit enter (you only need to do this once)
If the lights start flashing randomly, you did it right! The machine is now waiting for you to enter one of the 62 easter egg codes
If you enter a bad code, the lights turn red for a few seconds
But a good code makes the lights turn green! Enter one of the codes and press enter.
There are many codes, but you only need to do one of them to get the achievement. I will not be posting the other codes, by request.
Now, after you enter a valid code, go back to Hunrath (the main area) and go to Farley's house (the white one with the picket fence)
Turn on the projector, and the image will be displayed, and you should get the achievement