Dual booting means that you get a choice at startup to run either operating system; it sounds like that's what you have going on. It's easy to get the official AMD drivers for the Radeon HD4200 from this point:
Boot into Ubuntu.
From the Ubuntu desktop, click on System>Administration>Additional Drivers.
Once it finishes searching, Ubuntu will present you with a list of proprietary drivers. Click on the ATI driver and then click "Activate".
Reboot the computer and go back into Ubuntu.
At this point, you should be running the official AMD drivers!