Grub not showing on startup for Windows 8.1 Ubuntu 13.10 Dual boot

First, turn off the Windows 8 Fast Startup feature, as described here. That feature causes filesystem corruption, which can in turn cause odd boot issues. There's a good chance that this issue caused your problem initially. Disabling Fast Startup might or might not cause GRUB to reappear the next time you reboot.

If that doesn't fix the problem, you should try re-registering GRUB with your firmware's NVRAM. The way to do this that's most reliable is to use bcdedit in Windows. Open an Administrator Command Prompt window and type:

bcdedit /set {bootmgr} path \EFI\ubuntu\grubx64.efi

If you're booting with Secure Boot active, change grubx64.efi to shim.efi (or maybe it's shimx64.efi; in Linux, check the contents of /boot/efi/EFI/ubuntu to see what's there). Doing this in Windows is more reliable than other methods because some users have reported that some versions of Windows repeatedly re-register themselves as the default boot loader if the default is set outside of Windows. Although this is rare, it's consistent with what you're seeing, so using Windows for this task may be necessary.

An easier way to do it, but potentially less reliable in your case, is to use Boot Repair. This tool automatically fixes certain types of boot problems, and it's likely to work for you. The biggest risk is that if an overzealous Windows caused GRUB to become inaccessible to begin with, the Boot Repair change won't work.

If you continue to have problems, post back with details. (If necessary, edit your original post and comment on this one to alert me to the change.)

If you are usinge Windows Power Shell, you have to use Quotemarks:

bcdedit /set "{bootmgr}" path \EFI\ubuntu\grubx64.efi

I have one piece of advice. I followed the first answer process, but GRUB still was not showing up.

I followed the Fast startup off and the bcedit command, but it doesn't work. I was ready to use Boot-Repair in a Linux live USB, booting from BIOS, and I noticed a feature in 'Boot Manager' options that still showed 'Fast Boot' enabled, even when Windows said it was not enabled.

I turned it off, rebooted and there it is, as fast as always the GRUB interface!

I hope it helps before someone tries to reinstall Ubuntu or use Boot-Repair.

Edit 1: The 'Boot Manager' tab is the first tab in my BIOS, there is a list of devices from where I can choose to boot (hard drive, CD, USB...) Above that there are two options 'Fast Boot' and another that I don't remember. I hope it clarifies the ambiguous context of my answer.