How big would a UI that covers the entire screen need to be?

[HolyBanana]

Basically the size of the game screen.

I know I just can't load up the game screen. It locks up my computer.

It locks up your computer?
Try using google chrome.
If you are having issues loading the Java applet in-browser on GC, go to the Tools drop down and click on View Background Pages (...). Then end the Java process and try to load the applet again.