I just prove to them that I'm a stellar animal owner and tenant; I have references, the dogs' certificates from training classes, vaccine & lisc info, my cats are indoor only. Even before my pit bull died, I had some awesome landlords who were willing to give us a chance (who then become new references). Never lie about anything, and if your animal messes something up, fix it, don't leave it for the landlord. When renting, I behave as if I'm a guest with special privileges, basically.
As far as getting a new pet, I just talk to them. I give them the reasons why I want to add an animal, I don't lie about how big it will be or anything, and I respectfully leave the decision up to them (it's their property). To date, I've not had anyone say 'no', and all four of my current pets were acquired while I was renting. I've even been allowed to have some fosters.