Java is properly installed in my Ubuntu 12.04 and the PATH variable is set in the .bashrc
file.
The command java -version
gives an expected out
Install eclipse via the package manager and you shouldn't have this problem.
You've installed eclipse manually in your home directory and eclipse is trying to find a jre where it was installed, but you probably didn't download a version of eclipse that comes with its own jdk. As such it needs a little more help to work.
Like I said above, delete your current installation and install via the package manager.