YouTo be able to use your graphics card to do GPUGP, the easiest way to learn is to use OpenCL.
In order to use it with your AMD card, you'll have to install the AMD APP SDK. On their download page they have a package that is compatible with Ubuntu 10.04, but ought to work OK on 10.10 also.
UPDATE: current version (as of Q4 2012) requires Ubuntu 11.04 or newer,
On the documentation page you can find tutorials.