Here is one way for the Raspbian armhf (hardware floating point) using Iceweasel browser, OpenJDK and the IcedTea plugin:
Step 1: Install the packages (I am installing the JDK but you could also use the JRE if you are not planning on doing any of your own Java development)
sudo apt-get install openjdk-7-jdk
sudo apt-get install iceweasel
sudo apt-get install icedtea-7-plugin
Step 2: Add the IcedTea plugin to Iceweasel
#cd to the .mozilla directory in your home directory
cd ~/.mozilla

#create a plugins directory where Iceweasel will look for plugins
mkdir plugins
cd plugins

#create a softlink to the IcedTea plugin in this directory
ln -s /usr/lib/jvm/java-7-openjdk-armhf/jre/lib/ .
Note - this will run your applet with Hotspot Zero i.e. no Just in Time (JIT) compilation so it will run like molasses :(
There is an alternative vm JamVM and once I work out how to get it working in the browser I will update this article.