What you need:

• a PSP with 3.52 M33-3 or higher. (Use this tutorial to update your psp to that firmware, if you don't already have it: 3.52 Installation Guide

• The files below:
  
  
  
  
  Installation:

First download the files above and extract them

1. Using the USB Cable, connect the PSP to your Computer.
2. From the 3.90M33 Easy Installer folder, run the Easy Installer (on your computer, of course). Select your psp type (fat or slim), and the PSP drive letter. You can find that by connecting your PSP to the computer via USB Connection, then going to My Computer. See the picture below. When all that is done, Click INSTALL. It Will now copy all the files you need to your PSP.  
3. On your PSP, disconnect from the computer. Navigate to the GAME menu, and highlight the 3.90Installer (it should be the first one). Press X to start. Press X again when it prompts you to start the installation. Follow the instructions on the updater. After the update, press X or O to manually reboot. It is important to do this last step to avoid a brick
4. Congratulations. You now have a 3.90 M33 Powered PSP.
5. Back to the computer: from the Update3 folder you extracted before, copy the UPDATE folder onto your PSP into PSP>GAME>. IT will ask you if you want to overwrite the current data, click YES. (So the EBOOT.PBP inside the UPDATE folder is in PSP>GAME>UPDATE).
6. Disconnect again, go to the GAME menu, and run the UPDATE3 Updater. Press X to launch it. Press X again to start the update.
7. Congratulations, you just updated your PSP.