This tutorial has been written with the assumption that you have the Raspberry Pi up and running with a working internet connection. For information on getting the Raspberry Pi set up, take a look here.
There are two ways that you can approach this. The first is installing from the command before launching the X environment, the second is from the terminal once you've run startx. To be able to capture the steps nicely I'll be capturing the steps via SSH, but the process won't differ from a standard Terminal prompt launched from the GUI.
Once you've identified how you're going to install Chromium (either from the Terminal, or when booting up before you run startx), you need to run the following installation command: sudo apt-get install chromium-browser. This will connect to the repository, download and install Chromium.
Once you've run that you'll be prompted to confirm the components to be installed - simply hit Y.
The installation should proceed finishing with a confirmation that one
There is a final step required to complete the installation. Generally required, although if you're performing this on a clean Debian installation, and installing this before starting up the GUI, you don't need to do this step. Simply run the following command: sudo chown pi:pi ~/.config. When running it nothing will happen - you sill simply be returned to the following line.
That should be it - simply browse to the Internet menu and click on the Chromium icon:
And Chromium should start up!