I think @OscarRyz's approach is a reasonable idea.
However, I think you should think hard whether you should try to do this. The problem is that painting log messages into a Java GUI is likely to be CPU intensive. This is likely to make your application logging slow, and is likely to perturb the timing of your application as a result.
(Not that your application should be timing sensitive. But if you do have timing related bugs, it is not helpful if changing logging levels etc causes the application to behave differently.)
My feeling is that embedding a fancy log viewer in your application is likely to be more trouble than it is worth ...