Minecraft on slackware
 

I've installed java JDK and JRE and I'm not quite sure where to go from here.

Hi and welcome to LQ.

First of all, do NOT explicitly install both packages. JRE will do.
See: http://docs.slackware.com/howtos:software:java

As for Minecraft, you can use Alien's slackbuild:
http://www.slackware.com/~alien/slackbuilds/minecraft/

Do you know the specifics of how to install because I ran the SlackBuild and I have no idea how to run the game, there is nothing on my desktop.

Here's how to use SlackBuild scripts:
http://slackbuilds.org/howto/

After correctly building and installing it, it should be in your PATH and menus. Which desktop environment/window manager do you use?

the easy way is to install jre and go to mincraft website and download the client jar file from there. download it to your desktop.. than run it from there or create a little sh script so you can run it. its the same as it is with minecraft server jar file. google shell be your friend for quicker answers.