As Jacob mentioned, the CentOS packages repo appears to only have PHP 5.3 available at the moment. But these commands seemed to work for me...
rpm -Uvh http://mirror.webtatic.com/yum/el6/latest.rpm
yum remove php-common # Need to remove this, otherwise it conflicts
yum install php56w
yum install php56w-mysql
yum install php56w-common
yum install php56w-pdo
yum install php56w-opcache
php --version # Verify version has been upgraded
You can alternatively use php54w or php55w if required.
CAUTION!
This may potentially break your website if it doesn't fully resolve all your dependencies, so you may need a couple of extra packages in some cases. See here for a list of other PHP 5.6 modules that are available.
If you encounter a problem and need to reset back to the default, you can use these commands:
sudo yum remove php56w
sudo yum remove php56w-common
sudo yum install php-common
sudo yum install php-mysql
sudo yum install php
(Thanks Fabrizio Bartolomucci)