Joomla 3.6 will prevent updating CMS through Extension Manager even further
Since Joomla 3.5 the Joomla update packages doesn't have joomla.xml file, which prevents uploading the package through Extensions > Manage > Install > Upload Package File option. Joomla developers wants you to use the official Joomla Updater component instead, which isn't a bad thing at all - this method is generally faster and safer, however if it won't work for you - read on for a workaround.
For Joomla 3.5 this could be overriden by copying over the joomla.xml file to the root of the package, however since the latest commit on Joomla's repository by Michael Babker and George Wilson this also will be blocked in the upcoming Joomla 3.6 release and should not be done, because it may cause the same errors which caused Joomla developers to abandon this path.
If for some reason you cannot use Joomla Updater component or prefer updating manually - the optimal method as suggested by Michael Babker is as follows: download the newest Joomla release and unpack over your existing Joomla files and folders overwriting them, then use Michael's script to do the "finalise" and "cleanup" steps from the update component - you can find this script here. Alternatively check out the official Joomla documentation here for other methods with detailed steps on how to update your Joomla manually.
Of course this doesn't change anything for Perfect Dashboard users, you will still be able to use our automatic updates system to manage your Joomla websites without any worries!