According to this article on Phoronix support for your graphics card was added to the open source AMD driver after the initial release of Ubuntu 16.04 (on which elementary is based).
You should be able to install AMDGPU Pro by following the instructions for Ubuntu 16.04 without issues.
You will have a much better experience with the open source drivers than with AMDGPU Pro (things are less likely to break).
However if you want to install AMDGPU Pro you will still need to update the open source drivers as below (as you sill need a kernel update to support your graphics card).
Ubuntu supply updated drivers for their LTS releases to enable new hardware (as each release is supported for 5 years). Instructions to install these drivers can be found here.
To install updated drivers run the following command in a terminal and restart your system after it completes:
sudo apt-get install --install-recommends linux-generic-hwe-16.04 xserver-xorg-hwe-16.04