Open up the Game Boy using a Triwing screwdriver by removing the 6 screws on the back (2 inside the battery compartment).
As you open up the console you will see a large ribbon cable connecting the two halves. Grab it firmly and pull it out of the white connector on the bottom half of the console.
Put the front shell aside, and on the back shell, unscrew the 4 screws (2 on the top PCB, 2 on the audio PCB) and remove the back PCB assembly.
Using a soldering iron, warm up the pads of the 4 wires connecting the original power board to the wire harness (the one with 3 green wires and 1 red wire), and pull the original power board away from the wires.
Cut the grey plastic between each wire that keeps them together, so that each wire is free to move independently.
Solder the wires to the CleanPower board, in order so the first green wire goes to VIN, then the second green wire to GND, the third to LCD and the red wire to the last pad.
Place the PCB back into the back case, making sure to put the CleanPower down the side of the battery compartment where the original was, and screw the PCB assembly back together.