For future ref, this is how you reinstall OS X using a time machine backup:
1. Plug in your TM drive to the Mac.
2. Boot to recovery environment (CMD-R). If the system can't find a recovery partition, it will fall back to booting from Apple's servers. If this happens you will see an Earth globe with a progress bar below it. The bar represents the download progress of the boot/recovery image from Apple's servers.
3. When available in the recovery environment, open disk utility, and use it to repartition and reformat the internal disk (this may not be possible if it booted off a recovery partition on the internal drive, as there will be a lock on the drive, preventing repartitioning. If that's the case, then just reformat "Macintosh HD" (or whatever the OS X partition is called.)
4. Use the recovery menu/wizard to restore system from TM Backup.
https://support.apple.com/kb/PH18848