Make sure that you have Java (JRE) installed on your system
Download the Linux binary zip (no JRE) from http://www.soapui.org/
Unzip it into a preferable directory such as your home folder or /opt
Make sure that you have proper permissions on the unziped soapUI folder
Go into the folder and run
$ bin/soapui.sh (open source) or
$ bin/soapui-pro.sh (Pro)
Once you've downloaded the installer the installation is pretty strait forward. Start it by just double-clicking on it. After preparing the installation which takes only a few seconds you'll see the starting screen:
Just click Next to continue. You'll be asked to accept our license agreement
After accepting and clicking Next you'll be able to select the destination folder which by default is set to home/User/SmartBear/SoapUI-5.2.0 but can be changed.
Next step gives you opportunity to optionally include, as additional components, SoapUI source files, Hermes installation and tutorials.
If you've chosen to install Hermes, next you'll get the Hermes license agreement
If you have chosen to install tutorials, you will then be prompted to specify the tutorials location.
After that, specify the location where SoapUI will place symbolic links to.
After that a window appears where you'll be asked to select if desktop icon should be created.
Finally click Next to start the installation.
After couple of minutes at most, the installation should be finished and you should see the next screen.
Now you are ready to use it. Good luck