Install without the Play Store
On a Fire TV Stick or a box that has no Google Play, you can install Tevegram yourself. It takes about five minutes and you only do it once. If your device does have the Play Store, use it: updates then arrive on their own.
Fire TV Stick and Fire TV
Allow installs from unknown sources. Settings, My Fire TV, Developer options, then turn on "Apps from Unknown Sources".
If you do not see Developer options: Settings, My Fire TV, About, then press the device name seven times. They appear.
Install "Downloader" from the Amazon Appstore. It is free, and it is the usual way to install an app by its address.
Open Downloader and type tevegram.tv/apk in the address field, then Go.
No https, no www, nothing else. The file is about 18 MB and downloads in a few seconds.
Install, then Open. Fire TV asks for confirmation once, then Tevegram is in your apps.
Sign in with the QR code shown on the TV, using Telegram on your phone: Settings, Devices, Link Desktop Device.
Other boxes without Google Play
Android TV boxes running AOSP, and projectors sold without Google services, follow the same idea: allow unknown sources in the security settings, then fetch tevegram.tv/apk with any browser or downloader the device has. Some boxes ship a file manager that can open the downloaded file directly.
Updates
There is no automatic update on this route: nothing is watching over the app. When a new version comes out, repeat the Downloader step with the same address. Installing over the existing copy keeps your account and your library.