Use the YouTube app. Within the YouTube app under your account you can find purchases. Within purchases are playlists of seasons of TV shows which you can queue up to play in a row.
Netflix still does it better as it checks on you occasionally so you don't binge every episode.
You may need to install an app to play these movies. As mentioned by eldarerathis, the Play Movies app doesn't play local content, only shows/films downloaded via Google Play.
MX Player, VLC and MoboPlayer are popular media playing apps. Personally I use MX Player.
A lot of people seem to be having problems with Google Play Movies. If you have a look on the play store page there are a lot of people complaining that the app is buggy.
Lots of people have fixed issues by uninstalling and reinstalling, it may not help but it's worth a try.
I had a similar problem on my Lenovo P70A on Android 4.4 and 5.x as well.
The solution for me was to manually create the folder "com.google.android.music" under the "/Android/data" folder on my external SD card, which is where Google Play Music normally stores the downloaded (offline) music. Once I've done this and force closed Google Play Music, starting ...
You downloaded it via Google Play Movies, correct?
The file is encrypted with DRM locked to a combination of your account AND the device used to download it. No way you're going to get that to play on anything else. You'll definitely need to redownload it it to watch it on the 10.
I just bought and downloaded the 90 minute Google "AlphaGo" documentary movie in HD. It was about 2 GB in size.
It was saved on the SD card in this folder (I obscured the exact name with *):
In that folder were 156 files with names like this made-up format: