Hire a Lawyer and bargain for a Deferred Sentence. This means you would plead guilty, but the plea will merely be a notation in the Judge's file (and no further). The judge will give you a few classes or instructions, and will offer a future court date when you should return.
During the time before you return to court, you will need to ensure adherence to all of the Judge's rules and prohibitions. If you comply with them, you can get the charge dismissed from you record, allowing an Expungement later.
If you have specific questions, call me at 512-216-3211 (weekends okay). This advice is merely that; advice; what a judge tells you in court may differ depending on your case, facts, and history.
It sounds like you have already been found guilty (even though you plead no contest) if you simply paid a fine. You need to contact a local criminal defense attorney ASAP to see if you still can file a Motion for New Trial (within 30 days of your plea).
If you have already plead guilty and have missed the window for filing a Motion for New Trial, then unfortunately this is a final conviction on your record for the rest of your life. Austin is pretty bad about reporting Class Cs to DPS, so that might help you, but you can't get this sealed or expunged if it's a final conviction.
If you pleaded "no contest," then there is no way to clear your record of this.
If the PI was in the City of Austin, then you can probably move for and receive a new trial if it has been 90 days since you pleaded "no contest." You can call me (or Paul; he's a smart guy, too) to discuss this.
The record of your PI could arise from two sources: the court record with the City of Austin, and the Travis County Jail, if you were arrested. Either or both of these could show up on a background check.
In this economy, any scrape with law enforcement could affect your employment possibilities. You need to contact a lawyer and start sorting this out.
This answer is based upon the limited advice presented in the question, and is not intended to be legal advice, nor create an attorney-client relationship. For more information on your particular issue, please contact a local attorney.