You can't boot the Ubuntu installation that is currently installed on the hard drive, so you have three alternative ways of accessing the data on the hard drive. They are listed in descending order of speed.
- Install another Ubuntu OS as a dual boot on the same hard drive. Install TestDisk recovery application in the newly installed Ubuntu OS.
- Buy an inexpensive external hard drive enclosure and mount the old hard drive in it.
- Boot from an Ubuntu live USB, and install TestDisk on Ubuntu which is running from the USB as a live session.
TestDisk is a good software to use for recovering files. TestDisk is relatively fast and usually does a good job. If TestDisk can't recover all the files, there are other recovery applications described in How to recover deleted files? that work more slowly than TestDisk and may do a more thorough job of recovering files that can't be recovered by TestDisk.