The first step is to be sure what your record shows. If you have records from the sentencing date or from probation that has your conviction(s) listed, then you have completed the first step. If not, the clerk's office of the court you were in can provide this information.
Was it a felony or misdemeanor?
If you were convicted of a misdemeanor then in most cases you are automatically entitled to have your record cleared. Some misdemeanor's and all felonies require the court to make a "good cause" finding if you are eligible. A competent attorney will put together facts and argument for the court showing why your case justifies relief.
If you were charged with a felony, can it be reduced to a misdemeanor ?
Some felonies could have been charged as misdemeanors in the first place, like some assaults and some theft related charges. If that is the situation in your case, you may be eligible to have your felony record reduced to a misdemeanor record. This has wide ranging effects on employment opportunities, professional licenses and insurance rates.
Can I do this without a lawyer?
A lot depends on the charges and the county in which you were convicted. Some counties have made it quite simple for people convicted of misdemeanors to seek this relief on their own, others require motions and argument better left to a qualified lawyer. In some instances, like possession of a small amount of marijuana for personal use, the court expunges the conviction after two years without you having to do anything.