How do I update the firmware on an Apple TV 2nd generation?

You cannot update firmware however you can update TV OS. I would recommend looking over https://support.apple.com/en-us/HT202157 and follow:

1. Go to Settings > General > Software Updates and select Update Software.
2. If there's an update, download and install it.
3. Wait for your Apple TV to download the update. Don't disconnect or unplug your Apple TV until the update completes.

The last update for the Apple TV (2nd generation) was in 2014 (v6.2.1). No newer software is available for that model.