Technically, it is not a license, unless you plan to become a dealer. It is an application for tax paid transfer of an NFA firearm. The Form 4 is not available for down loading, but if you want to see what is involved and look at the form, write or call:
ATF Distribution Center
P.O. Box 5950
Springfield, VA 22150-5950
Phone: (703) 455-7801
and ask for a couple of copies Form 5320.4 (or just Form 4) and the fingerprint cards as well. No obligation, no charge.
Again technically, the Chief LEO sign off is not a background check; BATF does that. It is a certificate that your ownership of the firearm or item in question would not violate state or local law. Some CLEO's unfortunately have used it to prevent NFA item ownership in their jurisdictions; I have even heard of cases where a CLEO has demanded a bribe for signing off.