If you wish the desktop Spotify app to connect to your AirPort Express Wi-Fi network (assuming your external speakers are wired to it), do this:
In your very top menu bar, go to your sound options:
Speaker -> Choose your Airport Express
If that doesn't work, try this:
1. Connect your computer to the same WiFi as your AirPlay device (AirPort Express, etc.).
2. Hold down Alt Apple option key on your keyboard.
3. Click the speaker icon in your menu bar.
4. Select the Airplay device
If at first you don't succeed, reboot everything -- I mean it! Everything: desktop/laptop, airport express, etc. Everything.