UPDATE:
Step 2 is no longer needed, provided that you use a newer version of the backup managers mentioned in step 3.

First I would like to make the following statement:

I don’t condone the use of pirated video games. Support those that makes the games. Use this hack to run homebrew and your own legitimate backups.
Hacking your PS3 might be illegal at your choice of location.
I am in no way, shape or form responsible for what you do with your machine.
Continue on your own risk, and have fun.
Now when we have that out of the way this is what you need to pull this off:

PS3, fat or slim doesn’t matter.
USB drive with at least 180MB free space.
Step 1 – Update the firmware
First, make sure that you have the 3.55 original firmware (ofw) installed on your machine. This hack might work with other firmwares but I haven’t tested that myself so I give no guarantees. Now let’s download a prepatched (you could do it yourself, if you google it you will most likely find help on how to do this) 3.55 custom firmware (cfw) by kmeaw (there are other cfws like GeoHot’s, google for it if you want to try it instead), start off with downloading this file (mirrors). Unpack the RAR file and you will find a file called PS3UPDAT.PUP, create the following folder structure on the USB drive /PS3/UPDATE/ and copy the PS3UPDATE.PUP file to that location.

Put the USB drive in the PS3 and do an update via XMB. If you for some reason get an error that says something like “You already have the latest firmware installed” you will have to do this in recovery mode (read this). If the update was a success, go to step 2.

Step 2 – Installing the patch
Next we need to install something called a lv2 patch on our PS3. This patch needs to be run, once every reboot, before starting any of the backup managers for them to work properly. The lv2 patch is a package file and is installed via XMB. Download the lv2 patch here. Simply put this in the root of the USB drive and select Install Package in the menu on the PS3.

Step 3 – Installing backup managers
The backup manager is the software that allows you to start your homebrew and/or backuped games. I have a favorite manager called MultiMan (v1.13.04) (base package, mega package or another mega package if you use another firmware than the one I linked to such as GeoHot’s). Simply put the pkg file on the root of the USB drive and install it from the Install package in the menu.
Other backup managers that I think are worth to mention are:
Gaia Manager, the latest version (v2.04) doesn’t require to run the lv2 patch before running the manager. Download here.
Rogero Manager (v7.9b) for kmeaw (normal version, stealth version), other firmware (normal version, stealth version). If you play online, use the stealth version to lower the risk of your account being banned, don’t use this as a 100% guarantee that you will be safe though. The kmeaw version has the lv2 patch included so there’s no need to run that beforehand, I don’t know if that’s the case for the other one.

Step 4 – Using backup managers
All of the backup managers linked to above comes with a built-in ftp server which allows you to upload and download files directly to your PS3. To do this you need an FTP client such as FileZilla or whatever floats your boat, I personally use FTPRush which recently has become freeware. When you have started the backup manager of your choice you should see an IP address (e.g. 192.168.0.199) on the screen. This is the unique address to your PS3. Open up your FTP client and connect to that address, leave the port as 21 (default FTP port), select anonymous login and if possible uncheck Passive mode and/or passive transfers. If successful you should see some folders and especially one called dev_hdd0 this is the internal hard drive of the PS3, inside that one you’ll find a bunch of other folders but most importantly GAMES, GAMEZ and/or BRRips depending on where you told the backup manager to look for games.

Backup a game by starting the manager, insert the game disc, select it in the game menu and choose backup in the manager. This might take a while, depending on what game you are backuping, so sit back and relax and drink some coffee or something. To run a game from the manager, first make sure that you have any original PS3 game in the slot, then simply select it in the list and press X (if you’re using MultiMan, answer “No” when asked if you want to start the game from within the manager. That rarely works). After a few seconds you should return to XMB, now in the game menu browse until you find the disc icon. Then start the game as normal. This should work, if it doesn’t there might be some special procedure that has to be done for that specific game, try to google for it. You can also try running the game from another backup manager.

If you experience any problems let me know and I’ll try to help you out.