The first time you run the installer, it installs the latest version of Java if you need it installed.

After installing Java, you should reboot the Mac. If the installer doesn't automatically start up again after the reboot, you should try restarting the installer manually. The installer will then walk you through the rest of the installation.

Of course my knowledge is somewhat hypothetical, since LM won't install on my machine. If when you run the installer the 2nd timet (a) files appear in your trash and (b) the installer just quits and doesn't run, then your installation has failed silently. If so, try downloading the beta installer mentioned in this thread.

If the beta installer doesn't work, post complete information about your machine (model number, OS version, do you have a G3 upgrade card?). It won't do you any good but maybe next time around someone will figure out what the problem is...