I haven't seen this app mentioned, so excuse me if this is a duplicate.
I'm not really bothered by the Fire's lack of hardware volume buttons, but it is a bit clumsy to drag down the top bar and change the volume, especially in YouTube as it interrupts the video and there's some delay as the video reloads.
I saw this volume control app mentioned on XDA Developers (thanks to Lkr721993 wherever you are). It places a persistent and partially transparent slider on the edge of the screen. Sliding your finger up and down, or tapping, changes the volume without interfering with what you're doing.
You can turn the transparency all the way down so the slider is invisible but still works. I really like that you can make it completely unobtrusive. If the app you're using needs that area of the screen you can move the slider somewhere else.
There are other options I haven't tried yet, and it would be nice if a long click brought up the configuration screen rather than having to relaunch the app, but so far I'm very well pleased with it.
Unfortunately the app isn't available at Amazon. You'll need to do the install from Google market, backup the apk and transfer it to your Fire to install.
Volume control by RubberBigPepper