Since this is a 'Full Backup';

Login to cPanel and goto backup and Click 'Restore'.

Browse the backup file (tar.gz) and upload it.

Manually:

Upload tar.gz file to your home directory. (Do not upload to public_html)
After completed, Login to cPanel goto 'File Manager (Prefer Legacy)' goto to the home directory (Again, not public_html as you will see no files.) and find that backup*.tar.gz file and click on and on the side 'Extract'.

Now close the page that opened up after it has completed extraction and refresh the 'File Manager'.

Now you should see backup-*/ folder name.

Rename the public_html folder to public_html-OLD (or anything that is not there)

Click on the tiny folder and you can few the files inside there.

Your files are NOT stored in homedir/ folder, They're actually in homedir.tar.

Extract that file by clicking on and on the side click 'Extract'.

Now, refresh again, goto backup*/ and click on the littler folder homedir and now you should see all your files.

specifically, public_html.

Don't click the little folder this time click the public_html, text (as a link) and on the side click 'Copy' and should be able to move to your home directory.

You have your old public_html now in place.

Create a new database, new username and password.
Connect them together.

If your database is 'SMALL' upload it via phpMyAdmin.

If you're getting an error; If you have SSH use this command.

Goto your backup-*/ directory.
Goto to mysql directory, note the sql database you're looking for.
mysql -u username -p database_name < restorefile.sql
You will have to enter your password.

Alternatively, You will to use Bigdump http://www.ozerov.de/bigdump.php

Once imported.

You have to go into your public_html folder and find the location of your configuration file and update your new mysql database, username and password details.

and should be online;

Now login to admin panel and go update anything that has to do with your domain settings.

Alternatively, You can go into phpMyAdmin and use the 'Search Feature' and find your old domain and do some edits this way.

If you have no clue, I'm willing to help.