No Applications after Backup recovery
Not everything makes perfect sense, but I believe this set of instructions will help you along.

Review the entire articles below:


http://support.apple.com/kb/PH18872 - for the erase and reinstall  
http://support.apple.com/en-us/HT201250 - for setting up backups once the restore is complete


Outline of the process:


make sure the backup is safe and doesn't get erased. Also, be sure to not back up to this volume until your new OS has all the data you need since a new backup could cause older backups to be erased.
boot to recovery mode and open disk utility
erase the hard drive on your Mac
install a clean copy of Yosemite
make a test account (user name test-admin or something that isn't the name you used on the backup Mac)
update all software on the Mac
then connect the backup and use the Migration Assistant to move users and data. (you could move apps here too if you have any backed up)
reboot and log in to your "old account" instead of the test-admin account
verify the apps and data are there
then, connect and make a back up of the fixed Mac
delete the test-admin account and remove the home folder
enjoy your success