In .bashrc, just append the path to the directory containing the "java" executable to your PATH like you did with ~/bin. I'm not entirely sure what difficulty you're having, since you've already managed to do the same thing for your bash scripts!
You'll also want to set the JAVA_HOME variable. This can be done in your .bashrc, too. Set it to the directory which has been created after extracting the contents of the tar.gz file.