Information provided in the official Wordpress may be useful for your issue. Some of which I found fully documented over here:
http://wordpress.org/support/topic/your-php-installation-appears-to-be-missing-the-mysql-extension-which-is-require
Some of the possible solutions includes but not limits to:
Thanks for the reply, I found the resolution:
1) yum install -y mysql mysql-server - Ensure that MySQL and MySQL
server are installed
2) yum install -y php php-mysql - Ensure that PHP and the PHP MySQL
components are installed
3) Restart Apache
The same as:
Only one thing causes that error. Your PHP.INI doesn't have the mysql
line uncommented, so it's not loading the mysql module (or the module
is simply missing entirely).
Restart your webserver after you fix the PHP.INI file.
There are lots of forums and official support threads that refers to that last one as a final solution. But I also suggest you to install a MySQL Administration tool (or workbench, whatever you prefer) in order to make sure your MySQL installation is properly working. Some suggestions can be found in the answers provided right here: Are there any GUIs for Mysql? What is recommended?
First try the trick of the Wordpress Forums, if that does not work I suggest you to check if everything is ok on your MySQL installation and as an additional suggestion, you can find more useful and precise support on the Wordpress Forums just in case that the problem remains after checking that everything is ok in your server.
Good luck!