To dismiss a public defender, you should first make a list of the reasons he or she should no longer work for you. All the reasons should be related to your defense. Then file for a Marsden Motion which indicates that you are having a problem with your public defender. The motion will avail an opportunity to see a judge where you will present your reasons. Visit http://www.ehow.com/how_6674667_do-dismiss-public-defender_.html, for more tips.
I know someone who had to change his public defender. Are you doing it because you decided to pay for a lawyer or because you would like someone different? If you are paying for your lawyer you just tell her I found a new attorney but if you want a different public defender I think you have to ask that other public defender first to make sure they will take your case.