SOLVED
I tried to manually update Services Google Play & Google Play Store from the latest APK files.
This has stop the 'You must update the Service Google Play' message - but still the Google Play refuses to start 
I had the same problem when the Marshmallow update was released on Wednesday and I got over excited and chose to do the upgrade. Unfortunately, as soon as the upgrade finished and I got back into the main android UI, I kept getting a message that I needed to upgrade the Google Play Store and that Google Play Services kept failing. Of course, like other people have mentioned in this thread, updating the Play Store was impossible without Google Play Services working. 
After trying multiple things such as clearing the cache and data of both the Google Play Services and Google Play Store, and restarting my phone multiple times, it seemed there was no use. In fact, I think I was just getting spammed even more with the Google Play service failing to start. Not only was this annoying, but I was unable to update anything on my phone or download any new apps.
I checked if other people were having the same problems online, and long behold people were reporting the same symptoms. So I decided to go with the factory reset route and hoped for the best.
To do this I first hooked my Xperia Z5 phone up to the Sony's PC Companion and followed the steps to backup my phone's setting images, videos, call logs, contacts, etc. (I believe you can do the same on your phone in Settings>Backup&Reset>More>Manual backup, and do the backup on either an SD card or online account as well, but I chose the backup to PC route through USB).
After the backup finished, I went to my phone again and went again into Settings>Backup & reset>Factory Data Reset, and followed the instructions, confirming that I wanted to delete all the contents on my phone. Once the formatting finished, the phone restarted and I had to go through the same kind of setup process that you do when you get a new phone on Android.
Once I got back into the main UI the Google Play Services errors were gone. I then removed a few of the default apps that comes pre-installed with the phone, and started tweaking setting on the phone to my liking. Next I went to the Google Play store, and hit the menu button and went to Apps & Games > My apps & Games. Then I hit the "All" tab, and started downloading all the apps Google had kept track of that I had downloaded previously (I believe for this to work, you had to initially have setup Sync App Data, under Settings>Accounts>Google>Enable Sync App Data , before the hard reset).
Finally, after I reinstalled all my apps, I then hooked up my Xperia Z5 back to the computer, and again used the Sony companion app to restore all my previous settings, pictures, etc. from the backup I took before. Once the restore finished, I turned off my phone, and turned it back on (I did NOT 'hit' restart).
As annoying and lengthy this whole process was (at least 3+ hours), once I went back into the main UI after the restore, most my settings were back and all my files were restored. I had resign into most my accounts though (ex. email, Facebook, Twitter, Whatsapp, etc.). However, on the bright side all the errors were gone, the fingerprint sensor was working (after initial finger register setup), and I was on Android 6.0 - Marshmallow.
It terrible that this was necessary in the first place and I am very disappointed in Sony and Bell, to have released the software before testing it thoroughly (especially that they had so much time since Marshmallow released a while ago). 
I know this is not a good solution, however, I hope the steps above helps those who really want/need to get rid of the errors after the upgrade, and provides a solution that allows you to backup and restore all the files you have on your phone, along with fixing the issue. Otherwise, Sony/Bell will hopefully release a patch soon.For now it appears they removed the Marshmallow upgrade, and will be eventually rolling it again once the issue is fixed, according to their Twitter account.
TL: DR
Marshmallow upgrade released by Sony/Bell For Xperia Z5 and Premium broke Google Play Store and Services...
Only Fix I found (details above):
1) Make a backup of the content on your phone
2) Make sure you sync your app data with Google
3) Format your phone (DO NOT USE the Sony PC Companion to do this! Format using your phone explained above.)
4) Do the initial setup once your phone is wiped, including signing into Google
5) Let the UI load, and go to the Play Store and re-download all your apps you previously had (*Need to do this first before restore)
6) Use the backup you made previously to restore all the other content you had before
7) And presto! You now have a working Xperia Z5 with Marshmallow and all your files and settings back on your phone.
Sony has since taken down the Marshamllow update, good for any users who chose not to upgrade...unlike me. 
*Note: My Xperia Z5 does not have an SD card (I never add one, so I just store all my stuff on the intermal memory). Also the stuff I mention above is what worked for me, but I cannot guarantee that it will work the same for everyone else. I hope it does though!
Good luck, and sorry for the long post. It's my first one, but I wanted to help as much as I can since I know this is very frustrating.
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