If your background check includes a check of your Record of Arrest and Prosecution (RAP SHEET), then yes the DUI will show up even after 10 years. Generally, on the DMV Driver Record, convictions for traffic violations and DUI's do usually drop off. DUI's usually drop off after 10 years. So it depends on the extent of your background check and the employer.
How a DUI conviction affects employment
The answer above should not be relied upon as legal advice. The information provided above is based on insufficient facts and only speaks to a general opinion based on those insufficient facts. No warranty is provided that the answer is correct. No attorney-client relationship has been formed with me until a signed written contract is complete. For an official opinion, it is advised you seek legal counsel.