I downloaded the newest Java update yesterday, (Java
Environment 6 Update 1), and all of a sudden Azureus won't open. It loads,
displays the main window for a milisecond, then immediately quits. No error
messages, nothing. Updating to the newest version of Azureus solves
nothing. However, everything IS solved by going to Add/Remove Programs and
removing the latest Java update. Assuming this wasn't the first time you
ever installed Java, it should just roll back to the previous update and
everything should work fine. After removing Java 6-1, I also clicked the
"change" button next to Java 5-10, and just clicked next until it finished
(so basically changed nothing - I don't know if this mattered because I
didn't try it the other way.) Hope this helps some people.

Please refer to this url for incident detail
http://sourceforge.net/tracker/index.php?func=detail&aid=1708343&group_id=84122&atid=575154
I am keeping an eye on the forums at Azureus.
I have found interesting thing the All the Azureus users having this problem, have Alcohol 120 installed on their system.

Staffer from Azureus has to say that some of Alcohol 120 dlls somehow communicate with jre 6 update 1 and is creating this problem.

Thus to reproduce this issue probably it is better idea to install Alcohol 120 and then try to test Azureus and see if we can reproduce this issue.
I tried Alcohol try-n-buy version from their website without success to reproduce the issue.