tvOS has no Safari and no Files app, so a .mobileconfig can't be opened directly on the device. Instead, Apple TV downloads profiles from a URL through a hidden menu: open Settings → General → Privacy & Security, highlight Share Apple TV Analytics, press Play/Pause on the remote and choose Add Profile. The steps below walk through it in detail.
Apple TV can only fetch profiles from a link — there is no way to open a local file. Put the .mobileconfig on any server the device can reach. If it's a debugging-proxy profile, ProxyBeam generates it and hosts it on your Mac automatically — the link is copied to your clipboard.
On the Apple TV, go to Settings → General → Privacy & Security. Highlight Share Apple TV Analytics — but don't click it. Press Play/Pause on the remote — a menu with Add Profile appears.
Choose Add Profile, enter the link and confirm the installation.
Go to Settings → General → About → Certificate Trust and enable trust for the installed certificate. Proxy profiles need this step — without it, HTTPS traffic can't be decrypted.
Installed profiles live in Settings → General → VPN & Device Management. Keep one proxy profile at a time and delete it when you're done.