If I pay the back property taxes on some property in the state of Florida, can I claim the property as my own?

No. All you will be doing is paying the taxes for the owner. To take over a property for back taxes you would have to purchase it at a tax sale.

Each county in Fl will have a tax sale once a year for property that is normally 2 or 3 years deliquent in payment. And even after buying it at the tax sale, you will have to wait the redemption period of the state requireements before you can do anything with it.