How do you access the Zelda castle?
Once you get the Paraglider (or perhaps before, I don't know much about glitches or speedrunning tricks), you can head straight to Hyrule Castle, either through the front door or from the back. There will be Guardians in your way and you might get blown up a lot, but it is possible to walk straight up to the castle from the Great Plateau without doing anything in between.
To start the battle with Calamity Ganon, you must make your way into the castle proper. You'll find an area named "Sanctum" and once you step in, you'll begin the battle with Calamity Ganon.