First, if you are going for conforming financing, this means Fannie Mae & Freddie Mac loan programs, they have a lower DTI threshold than FHA financing. Fannie Mae's system can approve up to a 49.99% DTI, and Freddie Mac's system can approve up to a 54.99% DTI, but with a 625 score and 5% down - it's very unlikely your DTI would be approved to that level. With 100+ more points to your score, or with 20% down, it'd be in the "possible" category.
Further, when less than 20% down and Fannie or Freddie financing, a private mortgage insurance (PMI) policy is required, and right now no PMI companies are insuring loans over a 45% total DTI. Also, you may or may not be aware, but deferred student loans do not have their payments excluded with Fannie or Freddie loan programs - you either need to get something from the creditor stating how much the anticipated payment will be, or else lenders hit you with a monthly payment equal to 2% of the balance (which is a ton higher than what your actual payments would be). So with Fannie & Freddie, your DTI would be even higher because of that.
But with Fannie Mae financing, if you are not living in the home you will be renting out (like if it's an existing property you already owned) then you can put a renter in there, and with a signed lease and evidence of security deposit being paid, 75% of the amount off the rental agreement can help you qualify, it'll help "offset" that home's mortgage payment. Freddie Mac requires a 2-year landlord history before they'll count income like that. If you are living in the home you would be renting out, then with Fannie Mae financing, in order to use the rental income to help qualify, you'd need to have 30% equity in the home documented by an appraisal.
With FHA financing, a debt ratio up to 56.99% can qualify, even with 625'ish scores, and no reserves, and just 3.5% down payment. FHA's automated underwriting system is very lenient compared to Fannie's & Freddie's. When it comes to rental income, FHA has similar guidelines to Fannie Mae, in which you can use a percentage of the rental income to help you qualify without having a 2-year landlord history. If the property you own, you've owned for awhile, and you are not living in it (to clarify, this means that you have a tax return filed claiming that home as an expense on Schedule E, not as an itemized interest deduction on Schedule A), then you can use 85% of the rental income to help offset it's housing payment, lowering your DTI. If you are living in the home, and would be converting it to a rental, then FHA requires you either have 25% equity OR you are moving to a new area not within locally recognized commuting distance because of employment, then that is the other way you can use the rental income to help you qualify.
FHA loans do not include the deferred student loans in your debt to income ratio as long as they'll be deferred for the first 12 months of the mortgage - this means you need something in writing from the creditor stating when the repayment date beings/deferment date ends. It can't be ambiguous like "6 months after graduation".
In my opinion you'll have an easier time qualifying if you can get your scores up to the 640 level, as it's still the "norm" for most FHA lenders minimum score requirements. If you have a credit card at 35% utilization, paying that down to a $0 balance should get your more points - 15 points? Tough to say. But with clean credit for 36 months, even with a 625 you should have some FHA financing options, we'll go down to that score with your DTI - but again with a 640 you'll have an easier time in underwriting.
If you need help, have more questions, or want more information, please let me know.
These checks will have to show up on a bank statement (preferrably in an account utilized only for the rental). They will have to show as deposited and cleared. The tenants on the lease will have to be arms length. This means no relatives.
Lenders will most likely ask to see these items as part of the mortgage pre qualification process.