How to Show the Boundaries of Postcodes on Google Maps? [Solved]

Google doesn't provide this information "out of the box", but it lets you import external data in KML format and overlay it on the map.

You can find the KML postcode data you need here: Download USA Postcode Outcode Boundaries 

Create a custom map using My Maps. To use My Maps, just visit Google Maps, sign in using your Google account, click the "hamburger" menu in the top left-hand corner and then select "My Maps" and then click "Create a Map".

Then, import the KML file using the instructions here: Import info to your map. Style as required using the layer's context menu.

Another way to show the boundaries of postcodes on Google maps?

Your best bet is to use the free postcode point data from ordnance survey open data. Make a shape from matching postcodes, and then do a voronoi map round those matching points. It's not perfect, but neither is the propriety Royal Mail one.