Would using the drive as an external drive and backing up the data into it before placing it into the ps3 work?

I don't understand your question.

I plug in the hard drive as an external drive then backup the data from the original drive into that external drive and then I place that drive into the ps3. Would it work?

I guess you can use your external drive to backup the hard drives data and then start to copy the files into the new hard drives. 

I want to know if the ps3 will recognize the backed up data if the drive is instaled as the primary ps3 drive.

It will recognized backed up save data and everything else but i'm not too sure about games seeing as your going to have to redownload your games. 

It will if its in Fat32 format...

1) Format your external hard drive as fat32
2) Put your hard drive in the ps3 and redownload all your games and maybe backup your save data and what not to a usb

Also make sure to get your ps3 to format the external hard drive before redownloading all the data cause it will make its own files and folders and etc...

If your ps3 is hacked it would be a different story... I guess you can clone your ps3 hard drive to your new external and that might work with ur one... Good Luck t

I am just trying to skip the restore step.