I am trying to install FreiChat.V.9.1 with Joomla_3.1.1-Stable-Full_Package. However, I only ever get the following message when I attempt to access http://your_website.com/freichat/installation/index.php (I have replaced your_website.com with my own domain name):

Internal Server Error
The server encountered an internal error or misconfiguration and was unable to complete your request.

Please contact the server administrator, support@webhost.com and inform them of the time the error occurred, and anything you might have done that may have caused the error.

More information about this error may be available in the server error log.

I have checked permissions are all 755 for all subfolder and files.

I made sure extension=php_pdo.dll and extension=php_pdo_mysql.dll are enabled in the PHP configuration of my host.

I don't have access to the logs so I wasn't able to check them for error messages.

Is there anything else I may be missing?

you will need to check the logs otherwise it is impossible to determine what the problem is. contact your hosting provider regarding the same.
also, try to copy the permissions of joomla files and folders.