JavaScript per se does not allow to simply read the system clipboard as this would be a huge security risk. Most browsers do, however, provide the means to achieve this (although it has to be enabled by the user). The API is, thus, browser dependend. For Firefox have a look at https://developer.mozilla.org/en-US/docs/Using_the_Clipboard.
To the best of my knowledge there is na GWT specific wrapper of this functionality. Thus, you need to work with GWT JSNI (http://www.gwtproject.org/doc/latest/DevGuideCodingBasicsJSNI.html)
