If you have an HTML file that contains the applet tag, you can start appletviewer using Runtime.exec or the ProcessBuilder class. The HTML file would be a parameter on the command line (just like you'd run appletviewer normally).
If you want it to be part of your application, you can the class that extends applet to a Frame (of JFrame), since Applet extends Panel. Call init(), and then start() and stop() as appropriate. Be aware that you lose some applet features when doing it like this (namely, parameters and the applet context).