Effectively you are asking how to get audio from an arbitrary source (could as easily be your old portable video game as your Echo Dot) to share the speakers with your computer.
Routing it through your computer is an option, you would need to use the computer audio systems mixer controls to enable this - how you would access these of course depends on the operating system running on the computer, in many traditional desktop systems it would be on the volume control panel or an "advanced" tab thereof.
Another option could be to buy a small 2-input audio mixer and combine the output of your computer with the output of the device to feed into the speaker. This would have the advantage of avoid and additional set of analog to digital and digital to analog conversions going into and out of the sound card, and it would also mean that your Echo's audio output would work when the computer is off.