The emulator will automatically detect the game and sync your achievements to your global profile. Manual Verification
If you cannot get an emulator or watcher to work, many players keep a "Manual Log."
Some platforms allow you to manually mark games as "Completed" for your own personal satisfaction and collection tracking.
Use emulators like RetroArch, DuckStation, or PCSX2. Login: Enter your credentials in the emulator settings.
Here is a comprehensive guide on how to manage achievements for games that aren't in your official library. Use a Steam Emulator
This is currently the most popular choice. It saves achievement data into a local .json or .txt file within the game folder.