Extend Wi-Fi on a non-Apple network by Ethernet with an Airport Express
The short answer is yes.

Apple has an excellent support document -   Wi-Fi base stations: Extending the range of your wireless network by adding additional Wi-Fi base stations.  

Basically, what you're looking to do is connect via the Ethernet ports one or more AirPort (Express, Extreme, or Capsule) to an existing router or gateway.  See the image below.



Once you've configured your AirPorts WiFi settings to match your existing WiFi network you will be able to stream to any of the AirPort devices you have setup.  The Airport Setup Guide has further details on this.


  If you connect AirPort Express to your stereo or powered speakers, you
  can use AirPlay to play music from iTunes on any computer or from an
  iOS device on your network.