It is actually quite easy. First of all, you need to be off probation. If you are not off probation but have completed everything and are at least halfway through the term, you should be able to make a motion to terminate probation early.
If you get off probation or are already off, all you need to do is pick up some paperwork from the court, pay a small fee, and submit the case for expungement.
Mr. Spiegel is correct. However I hope you don't think an expungement will get this off your driving record. It won't and it won't even clear your criminal record. A CA expungement will simply add a note to your record saying that your conviction was dismissed. The arrest and conviction will still be there.
In addition to the answers provided above, you should note that if you are granted the expungement on your DUI, the DUI will still be considered a "prior DUI" for the purposes of sentencing should you be convicted of a subsequent DUI within 10 years. However, there are benefits to getting the expungement, If you do not have any other convictions and are asked on a job application or other questionnaire whether you have ever been convicted of a crime, you can legally answer "no". There are a few limited exceptions to this benefit, however, and should consult with a qualified criminal defense attorney to explore your options further.
A roundup of the best tips and legal advice.