1. Make sure you own Dungeon Keeper Gold through GOG
2. Add KeeperFX to your GOG account, by 'buying' the free game.
3. Open GOG Galaxy, find KeeperFX
4. Click the purple 'Install' button, and follow the instructions
5. Click the purple 'Play' button inside GOG galaxy.

Note that you do not need to launch Dungeon Keeper Gold in any way. KeeperFX is like a fully standalone game, you just need to own DK Gold.

It doesn't work.
The install button is missing, whether with or without the game installed.

For the majority of people it does. It seems that for a small percentage of cases, games bought through GOG.com do not immediately appear in the users library. This is not limited to KeeperFX or mods in general, and not something we from KeeperFX can do anything about.

The GOG recommended approach is to go to your account and 'DOWNLOAD OFFLINE BACKUP GAME INSTALLERS'.
You can also contact GOG support and they will make sure the game will appear on your library.
A third option is to wait a while, eventually it seems to appear in GOG Galaxy properly anyway. And the final option is to just go to KeeperFX.net and use the web installer there.

I hope this helps.