If you rent from an individual then it would vary by case but if you rent from a property management firm or a Realtor they are bound by law to treat each person equally and therefore if they run credit reports and background checks on anyone it wouldn't matter if you paid three years of rent up front they would have to run the same check on you that they do for anyone else.
Best practice is to be completely HONEST on your application for rental. Certain considerations are made for certain actions. Unexpected medical issues, low overall scores, even bankruptcy usually can be covered by extra deposit or increased deposits. Every agent and/or agency must provide you with their credit criteria if you ask for it. Most companies post theirs online so you have easy, readily available access to it prior to making an offer. If you have had an eviction, make a payment plan with the owner, landlord or representative of the unit to satisfy your debt. Do it in writing and have it signed by the representative and then make small payments or whatever you can timely and regularly so that when the eviction shows up you have proof that you are meeting your obligations. Some if not all companies see this as positive action to a bad situation and will consider you for tenancy based on the payments being made.