After you have purchased a Minecraft account, go to account.mojang.com/me or minecraft.net/download and download the Minecraft game client. Place it anywhere you wish, then double click the icon to run it.
You will need to log in with your username and password (or email address and Mojang account password, if you have a Mojang account). Please make sure that you are connected to the internet, as Minecraft will automatically download additional game files the first time the program is run. After the initial download, if you cannot connect to the internet, you will be able to select an option to play offline.
Minecraft was created in Java, so it must also be installed to run the game. To install Java, go to java.com/ and follow the instructions. Please note that while we normally recommend updating to the newest version of Java, some users have reported compatibility issues with Java 7. Therefore, you may wish to install the latest version of Java 6. Please also be sure that you do not have multiple versions of Java installed on your computer, as this can sometimes cause the game not to run.
If you experience trouble downloading or running Minecraft, please visit our technical support resources.