Recover Deleted Data from Malfunctioning Hard Drive

Your response indicates that you can boot into the recovery partition so I am making that assumption for this answer.

If you can boot into the recovery partition (accomplished by holding down Command+R while your computer is booting), there are a couple things that may work. Either using Disk Utility to create a disk image (easy but time-consuming) or using the terminal to transfer files (hard if you're not comfortable with the command line but more likely to work).

To create a disk image:

1. Connect an external hard drive to your computer, then boot into recovery mode by holding down Command+R at boot until you see the Apple logo.
2. Choose a language, then select Disk Utility from the options you are presented with.
3. In Disk Utility, you should see a list of all disks connected to your computer on the left-hand side. Choose the one that says Macintosh HD (or whatever you call your HD).
4. Click the New Image button from the top of the window.
5. Change the Where field to your external hard drive but leave the other settings alone (Image Format: compressed, Encryption: none), then click Save.
6. It will take quite a while if it works but, once done, you will have a mountable disk image containing all of the files from your computer. Test it by opening it on a working computer before erasing your HD though (I've made that mistake before).

If that fails, you can use Terminal to transfer files from the recovery partition by following these steps:

1. Connect an external hard drive to your computer, then boot into recovery mode by holding down Command+R at boot until you see the Apple logo.
2. Choose a language, then click on the Utilities menu option from the menu bar and select Terminal.
3. Once in the terminal, you can use standard terminal commands to find and copy data from your internal drive to your external drive.

I can add more info on using the Finder to copy files if you need it but this is a useful primer.

You have two problems: FileVault whole disk encryption and the HFS+ filesystem. HFS+ is not difficult to deal with in Ubuntu, but I've seen no readily accessible tools to access a FileVault encrypted volume even when you know the password.

It'll be far easier to use a Mac to turn off FileVault. Disk Utility can unlock and decrypt the drive. The folks at the Apple Store might let you use one of the computers in the store to decrypt the drive, otherwise you might have to borrow a friends Mac for a while. It'd be best to use the same version of OS X or newer.