How much can you afford/are you looking to spend total per month on a rental (including any utilities). Base your decision on actual numbers, don’t guess.
Research the area you are looking to relocate to and see the average rental rates there. This will help you decide on whether it is in your budget.
How is your credit? Most renters will require or ask to see your credit score these days. You should run a credit check for yourself from a reputable credit source and print it out to show the renters when asked. It will save you time, and also impress the renter that you have thought out the rental process.
Have proof of employment on hand and if possible a reference person at you employment that the renters can call to verify your employment information.
If you have a car, will you require private parking, or is street parking ok? If you are looking for private parking, be prepared to pay a bit more on the rent.
How long do you need the rental for?
Are you a smoker? If so, are you ok with smoking outside the home?
Be prepared to sign a lease. If a lease is not required by the renter, ask for an informal written agreement that you and the landlord have agreed to the rental situation. Make sure it includes how much you will be paying per month, what security deposit you have paid, how long you will be staying for and what utilities, if any, you are responsible for.