It's really simple. Download Minecraft.jar and put it in whatever directory you want. This is where the file will live.
Next, go to the ubuntu software center and search for java - you can use OpenJava if you want.
Finally, right click your Minecraft.jar and look for the permissions tab. There should be an option to allow it to run as an executable.
To run it, use this command in terminal:
java -Xmx1024M -Xms512M -cp /path/to/Minecraft.jar net.minecraft.LauncherFrame &The & sign at the end will allow the program to run without leaving a terminal window running all the time. I leave it out personally because it makes me feel like a badass.