There are two official HandBrake PPAs,
The former contains stable releases, which are updated about once a year. These releases tend to be rather out-dated when their end-of-life is approaching. The current stable version (0.9.8) was released on 2012-07-18.
The latter contains nightly builds, which are updated daily (or nightly, as it were). These are of course less stable, and undocumented to boot, but they are good software nonetheless. Additionally, as the stable release ages, the developers tend to start recommending users to try the nightly builds instead.
To add one of these to your sources, simply run:
sudo add-apt-repository ppa:stebbins/handbrake-releases
sudo add-apt-repository ppa:stebbins/handbrake-git-snapshots
depending on which you want. To install HandBrake with the GUI, run:
sudo apt-get update
sudo apt-get install handbrake
Alternatively, if you would prefer the CLI (command-line interface) over the GUI, replace the last line with:
sudo apt-get install handbrake-cli