Download the desktop app
The desktop app is definitely the easy way to go. When it's your turn, it'll get the save file and put it in the Civilization V save directory automatically (It will be called "(GMR) Play This One!" when you go to load a hotseat game). When you've finished your turn, your save file is off to the next player. No hassle, no worries.
In order to download the desktop app you will need to sign in.