If you are using Ubuntu 15.04+, many issues are now solved, so there is no need to follow this guide except if you are using any Ubuntu version older than 15.04. Depending on the version you are using (12.04, 14.04, 14.10) you might need all or some of the steps provided in this answer.

If however, you are using 15.04+, rejoice!, in all tests I have done with 15.04+ there was no need to do any of the steps mentioned here, so enjoy Ubuntu in all of it‘s booting glory!.

For the rest, before explaining the steps to do it, I want to be clear that I have tried many ways of installing Ubuntu with versions older than 15.04 (Or any other distro for that matter) from within Windows 8. No luck. Microsoft Windows really created a big mess for all Linux distributions. If you have a pre-installed Windows 8 system, you will probably never be able to install Ubuntu or any other OS in the normal (LiveCD/LiveUSB) or Wubi way. This is because Windows 8 introduced several new features, of which 2 are:

UEFI which substitutes what we have known as the BIOS (an alternative to)

Secure Boot which prevents anything but the installed operating system, in this case Windows 8 from booting. This is no longer the case for Ubuntu since 12.04.2

On a further note I want to mention something about Secure Boot taken from the UEFI Wiki

SecureBoot

"Secure Boot" is a new UEFI feature that appeared in 2012, with Windows 8 preinstalled computers. Ubuntu supports this feature starting with 12.10 64 bit (see this article) and 12.04.2 64 bit, but as PCs implementing support for it have only become widespread at the end of 2012 it is not yet widely tested, so it‘s possible that you may encounter problems booting Ubuntu under Secure Boot.

IMPORTANT: If you do find a bug, please file a bug report against the shim package in Ubuntu, preferably using the command:

once you‘ve installed with Secure Boot disabled. As mentioned by slangasek:

It is not required to disable SecureBoot in the firmware to install Ubuntu on a Windows 8 machine. Ubuntu 12.04.2 and 12.10 are SecureBoot-compatible. Any machine that ships with the recommended Microsoft Third-Party Marketplace keys in firmware will be able to boot Ubuntu under SecureBoot. If there is any problem file a launchpad bug for the shim package.

I also want to notify that I spend this week testing 15.04 and had an excellent outcome. Of 12 Laptops ( 4 Toshibas, 3 HP & 5 Lenovo) where Windows 8.1 was pre-installed, on all cases, Ubuntu detected the Windows Boot Manager correctly, gave the option to install alongside Windows 8.1 (It actually said Install alongside Windows Boot Manager) and solved any issues that appeared on previous Ubuntu versions. I basically did not have to do anything else on this cases. This was with Secure Boot on and on an EFI enabled boot system.

This does not mean that in your cases it will work perfectly, it just means that on my cases, I had a 100% Ubuntu Perfect installation. Again, with 15.04 (On all cases it was 64-bit).

So with all of this said, installing Ubuntu via WUBI is not doable (not recommended at least from me), things like trying to boot for example with the Windows 8 bootloader are not possible right now unless something changes in the future (There are also some bugs related to this in launchpad, like the one mentioned by bcbc). Apart from this, WUBI does not work from within Windows 8 if you have UEFI with a GPT Partition (Not MS-DOS type partition). Pleas read for more information.

Not only that, but trying to install 32 bit Ubuntu is impossible. You need the 64 bit version for everything to work correctly. Read more about Wubi in Can I install Ubuntu inside Windows?

The following is a small guide to install Ubuntu with a Pre-Installed Windows 8 system. The steps HAVE TO BE done in the precise order I mention them here to get everything started. If a step is skipped or done before another, you will most likely end up with some of the problems mentioned at the bottom of this guide.

For the time, you need to do it via a LiveCD, LiveDVD or LiveUSB, assuming (actually requiring) you have the following points:

You are using a 64-bit version of at least Ubuntu 12.04.2. 32-bit versions will not work.

Your system came with Windows 8 pre-installed (And you do not want to delete it)

You are not installing Ubuntu inside of Windows 8 but rather alongside of it. Inside of it is impossible because it needs Wubi which is unsupported.

Your system has UEFI activated (And can not be disabled) with Secure Boot.

You have already created a free space for Ubuntu from within Windows 8 with at least 8 GB (I recommend to leave at least 20 GB or so, so you can test the hell out of it).

You made sure that you actually have free space left on the drive to create the needed partitions and you also made sure that you did not have all primary partitions used (In case of using an MS-DOS Scheme) because this will create a problem with the Ubuntu installer showing you only the "Replace Windows" option instead of the "Alongside Windows" option.

You know how to burn a LiveCD, LiveDVD or LiveUSB from within Windows 8. If not, look for Windows apps that can do that for you. I do mine in another PC with Ubuntu ^^.