0. Make sure you have "multiverse" repository enabled in your /etc/apt/sources.list by clicking on System -> Administration -> Software Sources. The checkbox "Software restricted by copyright or legal issues (multiverse)" has to be enabled.

1. Download the latest LimeWire from http://www9.limewire.com/download/LimeWireLinux.deb or if you prefer the beta is here: http://www9.limewire.com/beta/LimeWireLinux.deb

2. If you are using Gnome or KDE you should be able to double-click on the file. If not, you may have to manually install with dpkg.

There should be an entry in the programs menu if you are using KDE or Gnome. You will also be able to start limewire by typing "limewire" in the terminal.

Enjoy!