If you say that you feel "bored", you mean you feel disinterested or you have nothing to do.
If you say you feel "boring", you mean that you make other people feel disinterested or that they don't enjoy talking to you.
When speaking English, it sounds more natural to say "I'm bored." for the first one, and "I feel boring." or "I think I'm boring." for the second one.
Hope that helps! :)