Try following this (yes, its got some age to it; but the game hasn't changed much since then) tutorial:
Installing Java On Ubuntu for RuneScape By Renegades009
Step 1: After a freshly installed Ubuntu system open up Synaptic
Enter the password you created at install. Now once its open go to
Settings Then click Repositories and make sure all boxes are checked
in first tab.
Now go back to the main window and search java. Scroll down to S and
mark installation for sun-java6-jre and sun-java6-plugin or something
Don't forget to mark jre but I have that installed already. Once done
click apply and click ok to get dependencies. When done with install
restart browser and walla! RuneScape time!
Also if RuneScape still doesn't seem to work for you reopen synaptic
and install openjdk.
Please don't rip my guide or pictures without giving credit and link
back. Thanks enjoy!
Runescape is of the same caliber as Minecraft; which both use Java JRE as a game engine. Ubuntu comes with a default install of Java, but you often need to fine-tune the settings in Wine or download 3rd party repositories, to get them to work.