Income: to see if you have 2-3 time the amount of rent in your monthly income
credit check: if over 600 then you should be ok, depending on if there are bad debts
Criminal Background check: Looking more at felony and high class misdemeanors
Rental History check: if you have any late payments or broken leases.
Apartments have a similar criteria but there are many out there that accept certain items. The main issue is income, if you don't have the ability to declare income then you might be better staying where you are. Almost all places will need to see income to verify that you can pay rent.
If needing to be in another place then driving around looking at signs that just say "for rent" with no company on them or local websites would be your best bet.
This doesn't mean that a landlord will automatically reject someone because of having a record....most landlords are more interested in whether you have the ability to pay the rent (enough income) and have a good history of paying rent and other bills on time.
Some landlords will only ask for credit checks and job verification.