I would like to remove the Notification Center icon like I could in Yosemite.
I'd like to not see it in the menu bar (to the right of the spotlight icon).
So you can easily disable it with
launchctl unload -w /System/Library/LaunchAgents/com.apple.notificationcenterui.plist; killall NotificationCenter
You can remove the icon with
sudo rm /System/Library/CoreServices/SystemUIServer.app/Contents/Resources/menuitemNormal.pdf
but the blank space will remain. There is an app called Bartender I've seen a few people promoting to fix this issue.
To follow on @walshie4's answer above: The second command will not execute with El Capitan due to System Integrity Protection.
You must first boot into recovery mode, start a terminal and execute the following:
csrutil disable; reboot
Also note that after deleting the menuItemNormal.pdf file, the area is still a blank clickable space.