Make sure to have a java runtime installed (OpenJDK,sun-java), and execute the minecraft.jar with
java -jar /path/of/minecraft.jar
alternatively, you can right click the jar file and select to run it with your installed Java runtime.
You can even create an empty file and input the
java -jar /path/of/minecraft.jar
and you have created a shortcut!
On it's first run, or any time you force an update, minecraft will download all of the necessary files to /home/[user]/.minecraft. The game also stores you saves, stats, screenshots, etc. here. For consistency, I would recommend moving the minecraft.jar you would have downloaded into the .minecraft directory after you update the game.
May you mine many diamonds.