First, make sure you have Sun JAVA installed. Installing the JDK or the JRE is a sufficient way of doing this. The JDK might be best. Read here on how to install it: How to install the Sun Java JDK?
Then, from the Minecraft site, you will need to download the beta .jar file. Download it to your Desktop.
Then, in terminal, do the following:
chmod +x minecraft.jar
username with your username that you login to your computer with.
Then right click the Minecraft.jar file, and go into Properties. Under the "Open With" tab, choose the option button that says "Sun Java Runtime" or similar wording. Hit the "Close" button.
Finally, double-click the minecraft.jar file. It should now run the Minecraft client.