The information below is accurate. Leases are on a case-by-case basis. Each landlord will have the ability to decide if they will approve your lease application. Please know that they will likely check not only your credit report but also your criminal background report. Most landlords also call prior landlords to check your payment history. The rule of thumb regarding income is 3x your monthly rent in gross salary, so based on the income you indicated you should able to lease a home up to approximately $2200 per month. Of course, you may be able to find homes for less per month.
As for finding a home available June 1, you definitely need to be looking right now. If you don't already have a Realtor, I would recommend you hire one. They can send you a list of homes available, show you the one's you are interested in, and help you to submit the application with the application fees. Please know that you will need to pay the security deposit up front (which is usually equal to 1 months rent), and then you will have to pay the first month's rent upon move-in. As stated below, if your credit is really bad you may need to pay a double-deposit....but that will be on a case-by-case basis with the landlords.
Another item you may consider is to write a short note/letter to the prospective landlord telling them your specific situation. Sometimes that helps a landlord understand your position and they may be more willing to work with you based on your circumstances (depending on what those are).
Good luck! If you don't have a Realtor - our team would be happy to assist you. We help home buyers and renters in McKinney and the surrounding areas. Feel free to call if you have any questions at all.
Melissa Hailey - North Texas Top Team, Realtors
Coldwell Banker Jane Henry Realtors