If your graphics card is supported by Ubuntu, here's how to install the proprietary drivers.
Click the Unity menu button and type "Additional Drivers", and then click the icon labeled as such.
When the window opens, if your card is supported, it will list available proprietary drivers. Just highlight the latest one, or the one marked as tested, and click "Apply Changes" then reboot.
However, I don't think your card is supported. I have a Radeon HD 4200 and it's not supported any more. The problem is that most of the newer, supported versions of Ubuntu use an updated version of the Xorg display server, and AMD has stopped supporting its older cards in newer versions of the driver that are packaged with Ubuntu, and the older drivers don't work with the new Xorg server so you can't just go get those and use them. The only way to get proprietary driver support without buying one of the newer, supported cards is to downgrade several versions of Ubuntu.
I just confirmed, your card is not supported. I just went and downloaded the deb file for the proprietary drivers and dug through the documentation, and this is what I found.
Quote: "The AMD Proprietary Linux driver currently supports RADEON 8500 and later, as well as FireGL 8700 and later products. We do not currently plan to include support for any products earlier than this."