Lets' try checking theses items.
First, your system has two different audio output systems, but Windows can use only one at a time. But let's start by ensuring both systems are working.
The mobo has a Realtek audio output system that feeds signals to the set of 3 output jacks on the mobo back panel. See your mobo manual, p. 31. There you find an option to Enable or Disable the on-board audio, so ensure it is Enabled. After that go into Windows Device Manager and check to verify the Realtek Audio system is there. Then check that its device driver also is properly loaded, or maybe update that if necessary. Then go into Windows Sound controls and ensure that Windows is using the Realtek on-board system BOTH for default playback device and for default recording device.
That should ensure that the mobo sound system will send out sound on the back panel jacks. BUT there is another "trick" involved in Realtek systems. Before you can do this step, though, check that the mobo sound system configuration utility that was included in your mobo's CD disk has been loaded so that you have a Realtek system icon in the lower right corner of your screen. Use that to set up which configuration you are using - for example, using both front panel headphones and rear panel 5.1 sound outputs. Now, Realtek has an interesting system that allows you to re-define what sounds come out of what jacks, and it can cause problems. I'll describe how mine works, and yours will be similar. This is for using a speaker set from the rear jacks, and NOT for using the HDMI feed to a monitor. Arrange your system so you have access to the rear jacks and the speaker plugs, can see you monitor screen, and use your keyboard and mouse. UNplug all the speaker plugs. Now plug in ONE plug - say, the front left / right stereo plug to the light green jack. Watch your monitor screen for a pop-up window that says something has been plugged into a jack and suggest what that might be. If it is wrong, correct it and confirm. If it is right, do NOT ignore it! You MUST reply and confirm what really is plugged in there. If you leave it alone it will disappear and NOT send out any sound signals on that jack! Once that is done, proceed to the next plug - say, the front middle / subwoofer plug into the orange port. Same drill - ensure the correct item is identified and confirm. Continue with rear left / right speakers, maybe the headphone on the font panel, until all your output devices are attached and identified. Now try playing sound through those speakers. You can use the configuration tool to do tests and check polarity, etc.
OK, so that should (we hope!) get you sound from the mobo rear panel via speakers, etc. IF your monitor has input jacks for audio from a sound source you can connect a cable from the mobo rear jacks to the monitor. This MAY be only the left / right front stereo signals, so you would have to configure the Realtek system to do sound that way, not in 5.1 surround sound. If that works, you have sound on your monitor that way.
Now for the preferred method - using the HDMI feed instead. It is really difficult to feed sound signals from a mobo system into a Graphics card, so almost all of them have their own sound system on the card. THAT signal (usually only 2-chanel stereo) can be sent out on an HDMI cable to a monitor along with the video signals. A few such cards also provide audio output jacks as an alternative. NOTE that normally sound signals ARE included in the output from an HDMI port, but NOT on a standard DVI port. I have a video card that does actually add stereo sound to the output on its DVI port (not part of standard DVI), and I have to use a particular adapter that converts the DVI output to HDMI for use with a normal HDMI connection to the monitor. This adapter has extra connections to pass the non-standard audio from DVI onto the HDMI lines. However, if you are using a standard HDMI output port this is NOT an issue.
There is another possible problem, though. When you install a graphics card in your system it very often does an automatic installation of the device drivers for that card. What is not obvious is that there are at least TWO drivers for the two devices - graphics output, and sound output - and they BOTH need to be installed. So, in your system, look again in the Device Manager screen and see if there is a second audio output system named like your Radeon RX 480 card, and different from your Realtek system. If you cannot find it, Windows does not know it is there and cannot use it. So if there's no sign of it, use the tools from the top menus to re-scan your system for new hardware and see if that sound output device on the graphics card can be discovered. Once it is found, you need to ensure that the proper device driver for it also is installed. You may find it on the CD that came with your graphics card, or maybe you can find the latest on-line from the card maker of from AMD. I see in your post you had some difficulty with that step.
OK, IF you can get both the mobo Realtek audio system and the graphics card audio system installed AND with good device drivers, now you can try to switch to using the card's system. Use the Windows configuration tool to change to using that graphics card sound output device (probably do NOT change the default sound recording device). Disconnect any cables between the mobo rear audio jacks and the monitor, and try using the system to get sound to the monitor using the HDMI cable. You MIGHT have to find and use some utility for configuring that new audio system, but I doubt that.
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