Yes, a bench warrant (assuming it was entered in the LEIN) will show up on a background check. You should hire an attorney and have the attorney set a date/time for you to turn yourself in and get this taken care of asap.
No. A bench/arrest warrant will not show up on a background check. The LEIN network the other attorney refers to stands for Law Enforcement Information Network. It's only available to courts and law enforcement, hence it's name. A background check only shows public information. The State Police will only share public information in a background check given to prospective employers, temp agencies, etc. It's called an ICHAT file. Warrant information is not made public as it can be a safety concern for the police. Only arrests, charges, and convictions will be included in the report.
Mr. Lapres is correct, but Mr. Simkins is correct in noting that the record exists, and could come into play if you apply to work in law enforcement, prision system, etc. You really should get that warrant cleared up. You can call the court and see exactly what the issue is. If it is far away, a local attorney can be hired to find a resolution for you that might not involved having to return to the county where the warrant was issued. Warrants seem to live forever, and can trip you up at the most inconvenient times.
We do not have a client/attorney relationship until you make an appointment, we discuss your case face to face, I accept a retainer, and we explictly agree to enter into representation.