Same with the criminal charges.

Obviously, they will probably do a background check and credit check which will show you will be able to pay on time and that you don't have a criminal record - furthermore, meeting with them in person and getting to know each other a little bit better will help determine if you will be the best fit for that landlord and their property.

(specifically pertaining to income and criminal).

There must be a time limit, including for evictions and criminal.

No offense to anyone.You should be using QuickBooks to budget, forecast, cash flow, knowing how much you owe and own, when, to whom, where etc. and know your equity. 

The background check is comprehensive and includes credit, evictions, judgments, address history, sex offender, criminal background, etc.References will be checked.Rent is $910 per month, $910 security deposit.

Does anyone have experience renting to tenants with prior offenses?

There's no guarantee, but: most people who have jobs, no criminal record, good credit scores, and are reasonably nice people rarely turn into nightmare tenants. 

I have seen these escalate to criminal matters.

The good:Clean cut, well spoken, has 2 verified sources of income (one is a transportation business he owns) and meets our income qualifications, good payment history for the past 2 years according to credit report, good communication with me, jumped through all of my hoops.The bad:While the past 2 years are "clean"....Chapter 7 Bankruptcy in 2013, Eviction (says he co-signed with a girlfriend), 2 divorces (which led to bankruptcy) and a "failure to operate vehicle with license" on criminal report.