You cannot technically run Java on a Chromebook. Chrome no longer supports NPAPI (technology required for Java applets There is no way to install Java on ChromeOS. Google generally views Java apps as being unsecure and limited the functionality on Chromebooks. You can not run Java on Chrome OS, as the security design prohibits it (and Java is a known security threat, with many, many examples). If you are comfortable with the security issues introduced by Java, you will need to install Linux. Java support on Chrome for Linux was dropped as a result of Google phasing out the NPAPI plugin system. Chrome for Windows is supposed to follow the same path.
Install Java on Chromebook
Chrome devices have Flash support built-in, but they do not support Java or Silverlight. If you need Java, Silverlight, or other plug-in support, there are virtualization and remoting options you can use for Chrome devices. See the chapter on Remote Access and Virtualization in the Chrome Devices for Education Technical Planning Guide (PDF).
There are ways to get Java on a Chromebook, but these all require installing another operating system like Ubuntu via Crouton.