Here are two easy answers:
(1) Don't install Java 6 (the old version from Apple). Instead head to Oracle's website and install Java 8. Now head back to Minetcraft.net and download the Linux version of Minecraft. Extract the "Minecraft.jar", double click to start, and answer a few security questions.
(2) Wait. According to reddit post in December 2014, Mojang will bundle an appropriate Java with a future Minecraft launcher. This will run in 64 bit on 64 machines, and reduce potential Java version confusion. The download will increase in size from approximately 40mb to 60mb.
Mojang writes that Minecraft works best on Java 8 (and not Java 7!).