I agree with the other posts.
We all make mistakes. You made a mistake. You take the consequences. You learn from it. You become a better person. And you move on. I wouldn't bother writing the dean or the professor. Sorry to say, but they don't really care. They know the rules of the game, and it is all part of the job for them.
You shouldn't do it again... but it might not be as big a deal as you think it is. It isn't life ending. It may mean you need to do a little more work to get to grad school.
Apply for grad school if you ready. If you aren't then take whatever courses you need. If you need to take more courses, then take more courses. If you are rejected, find an adviser to help you build up your record and keep trying.
Yes, you have the chance to redeem yourself.