Most states won't do an extensive background check, but will do a general one to make sure neither party is already married, and if there is any warrants out for either one which is why there is a waiting period before issuing the license.
Do they do background checks in Iowa when applying for marriage license?
Generally speaking, comrade, they do not run a criminal background check for a marriage license in any state that I could find. Although it's a general consensus that they should.