When dealing with a Public Intoxication charge it is important to realize that an expunction of the arrest record will not be possible if the underlying charge is not appropriately dealt with. Pleading "No Contest" allows the judge to find you guilty and such a plea should not be given. It would be best for you to hire an attorney who can assist you in this matter and then assist you in obtaining the expunction. A Public Intoxication charge, while a class C misdemeanor, is a more serious offense than a traffic ticket and the consequences for not dealing with it appropriately can have long lasting effects.
It would be a bad idea to handle it yourself. This is a legal matter. If it's important for you to get it expunged, then hire an attorney. If you just want to fool around with it and see what you can do on your own, then by all means handle it yourself.
If you want to save money, you can cut your own hair - but you may not like the results! The same applies to practicing law!
This answer is for general purposes only, and is no subsitute for specific legal advice that would come from an attorney hired by you, having full knowledge regarding the facts and circumstances of your individual situation. This correspondence does not create an attorney-client relationship, and you should consult with one or more attorneys prior to acting on any of the information provided in this response.