Jerell,
Credit and rental history are probably your biggest worries - i.e. do they have the funds to pay and will they pay, and do they owe to their prior landlords. But for figuring out their income ratio, the Section 8 program will pay for a percentage of the rent, typically 70%. The remaining percentage, again typically 30%, is subject to the income ratios you require. So for a $2000 rent, $1400 might be covered by the program. The remaining $600 is the tenant responsibility. For a 3x requirement, they must have a verifiable income of $1800 (3 x $600). Any other rent items, such as pets, etc. must fit within those same calculations.
For Deposits, unless the program will also provide the Deposits (some do, some don't) they must have the appropriate funds. Be sure, before you offer the lease, that you know what is provided and what they need to bring to the table. Last Month's Rent is often not allowed under these programs because the program is guaranteeing the funds (or portion thereof).
Otherwise, stick to your qualifications and screening criteria. Be fair and be consistent. There are many good people under these programs, but likewise there are many looking to sell a sob story. If they do not meet the criteria, show a bad history of paying on time, or owe a prior landlord then it doesn't matter how nice they seem - they will continue being who they have been in the past.