Updating your PSP with custom firmware is really easy. Theres lot of ways to do it, but I believe this to be the simplest one, and its practically foolproof.
To install the firmware the only thing you need is Pandoras Box GUI and a Datel TOOL battery (or homemade Pandoras Battery). This program is hated by many PSP users because its so simple, but the only thing I’m bothered about is getting homebrew to run – if its too simple for you then go research it for yourself.
The first step is simple enough – insert your memcard into your PSP and connect to your computer in USB mode. I’ve never had any problems with anything but the recommendation is to use a PRO Duo memcard of at least 64mb. Backup your saves if you need to and then run the installer program. Everything in this step is explained from the program, so follow it making sure to reboot when told.
After the memcard has been formatted, insert your Pandoras battery and once the PSP has booted in safe mode (the power LED is on, but the screen is blank) insert your memcard, and follo it through. This will install your 1.5 firmware on its own.
Once you have this installed you will be homebrew capable. From here, theres not really much more to do. I reccommend upgrading to more recent custom firmwares. The best way to do this is to install 3.5 M33, and then upgrade this to 3.7 M33. These can be found quite easily on any PSP site.