In Windows 10, the best way I've found is to just use the Emoji shortcut: Windows Key + ; (semicolon)
Other option is to...
Use the Touch Keyboard:
Right-click on the Windows Taskbar and choose "Show Touch Keyboard Button".... which then places a little keyboard icon/button in the lower right side of the taskbar. Then you simply click that when you need to use emojis: There's an Emoji key right next to the spacebar on that virtual keyboard.
The On Screen Keyboard on the other hand (Windows Key + O ... didn't have an Emoji key- at least not for me. Not sure why that is... but I'm totally fine with the options mentioned above.