There may be a simple problem with the court where your conviction was entered in error or there may be a larger problem. Although potential employeres are only supposed to consider felony convictions, reality is that misdemeanors are routinely considered. You should consult with an attorney who can review the court records and make sure the conviction is not on your record. If your record shows that you were arrested and the case is dismissed, there may be a way to get that off your record as well. I highly recommend you use avvo to locate an experienced attorney in the county where your domestic violence case was pending. He or she will be able to find a way to help you.
Background checks are techincally not legal, and depending on how good they are, only then will they show up on your record. Only felonies can be expunged from your record, not misdemeanors. Most domestic violence charges are misdemeanors unless you have enough subsequent notices that they now have the affect of a felony. But most places of housing and employment only ask if you've been convicted of a felony, so in that regard it shouldn't hurt you that much if at all. Only law enforcement, and only then under limited circumstances, can access your criminal history.