Is this your first install of Frostwire?
Did your machine come with Lion installed, or did you upgrade from Snow Leopard?
Have you run all software updates. If not re-run and install, and then re-run if necc. until no updates are available.
Please go to Applications > Utilities > Java 0r Java Preferences and tell us what you see there.
Please Open Applications > Utilities > Console. Clear the console. Try to start Frostwire. Enter Frostwire into the Console string Filter and post the resulting log, if any.
I thought the Lion compatibility had been fixed .. it looks like possibly not. All information you can give will be very useful.
Please do not Private Message me with questions that can equally well be posted in the forums, and may also help others. Do use the forum search.