If you have a PSP Fat (regardless of motherboard version) or a Slim with ta-090v1/ta-088v2 or below with official firmware 5.03 or below, you can install custom firmware by using Easy ChickHen custom firmware installer. If you have a ta-088v3 motherboard, stop reading, this will only brick your PSP.
1. Connect your PSP to your computer.
2. Download Easy ChickHen custom firmware installer here.
3. Open Easy ChickHEn installer and enter the drive letter of your PSP.
4. Wait for the installer to finish copying files to your PSP.
5. Go to GAME section on your PSP and open 5.03 update. Proceed to step 7 if you're already running 5.03. Be sure that you have 100% battery and put the AC charger in. Wait for the process to finish.
6. Go to system settings>system information to check if you are now running 5.03.
7. Go to GAME section again and open PSPident 0.4.
8. A screen will appear with details telling you what kind of motherboard your PSP has and will create a screenshot on the root of your memory stick.
9. Make sure your motherboard is hackable(see image below). Don't proceed if you have a ta-088v3 motherboard.
10. Open HC Recovery Flasher.
11. Select "I agree".
12. Select "Install M33 CFW".
13. Select either of the two options.
14.Wait for the flashing to be finished.
15. Press any key to shutdown. Open your PSP and check if you're running 5.0m33-4.
16. Enjoy!
-credits to pspslimhacks.com