First you have to install Ubuntu and after having finished it, boot into the installed system. Now you can install the AMD drivers from the Additional Drivers tab. Alternatively you can install the proprietary drivers by using the terminal, open a terminal and execute the following commands:
sudo apt-get update
sudo apt-get install fglrx
sudo reboot
Update addressing your comment that the open source drivers prevent you from installing:
When booting from the installation media, select Try Ubuntu without installing, press the E key and add radeon.modeset=0 at the end of the linux line ... press F10 to boot. Install Ubuntu.
Reboot - select the Ubuntu menu entry, press the E key and add radeon.modeset=0 at the end of the linux line ... press F10 to boot. Install the proprietary AMD drivers as described before.
Important : Put a Space between the last character of the linux line and radeon.modeset=0.
After the proprietary AMD drivers are installed, adding the boot parameter is not necessary.
Note: fglrx will not be available in the repositories of Ubuntu 16.04 LTS, which will be released in a few days, so when you want to upgrade you may want to consider not to install these drivers. Ubuntu 16.04 most probably will offer the new amdgpu (replacement) drivers in the repositories.
As a workaround you can can deactivate the radeon open source drivers by adding the parameter to the /etc/default/grub file, so that the GRUB_CMDLINE_LINUX_DEFAULT="quiet splash" line reads -> GRUB_CMDLINE_LINUX_DEFAULT="quiet splash radeon.modeset=0". Save these changes you have made and update GRUB afterwards - open a terminal and execute: sudo update-grub.