After a DUI conviction having it removed from your record can be an option.
This is called expunging the charge.
Depending on your state there may be different factors that play a roles in having your DUI expunged, but there are some general guidelines you must meet in order to have your DUI expunged you should know:

• You were not sentenced to prison
• You are no longer on probation
• You fulfilled all of your DUI sentence (fines,DUI classes, etc.
  )
• You have no pending criminal cases

You should learn if you are eligible before spending too much time or effort thinking about getting a DUI removed from your record.
If you believe you are eligible and want to proceed, you should contact a DUI lawyer to assist with the process.
While the process may not be cheap, it is up to you to decide if it is worth it, however simply contacting a DUI lawyer with your specific circumstances will cost you nothing except a little time.
To make things easier for everyone before you meet with the lawyer some common things to bring are:

• Your criminal history,
• Any probation documentation
• Your driving record
• Employment history

If you contact a lawyer dealing with expungement cases in your area they will tell you exactly what information you need and where to go to get it Keep in mind having your DUI expunged does not always prevent the conviction from playing a role in future DUI sentencing.
For example, if you get your DUI expunged and then 3 years later get another DUI you will be sentenced as a second time DUI offender and not be eligible for expungement for a long time.