Until the county goes through the levy and tax sale procedure, there's no way to get ownership of the property. Just because you paid someone else's taxes gives you no rights.
You're free to go to the tax sale when it happens and bid. The opening bid will be the amount of taxes owed and some administrative costs. You need to have cash (or certified funds such as a cashiers check) to pay for it on the spot. You're not going to be given any credit for money you already paid (other than it decreased the opening bid).
Now lets say you go to the tax sale and nobody else bids on the property so you get it for the minimum bid. It's still not yours. There's a one year right of redemption which means the owner or anybody else with a lien (like the bank) can come back and pay you off and get the property (though they will pay you a 20% premium and reimburse you for any taxes you pay in the interim). After a year, you can foreclose on the property and extinguish the other liens. Or you can wait four years until the tax deed "ripens" and then you get ownership by default (even without having to specifically foreclose).
So if you've got a lot of money burning a hole in your pocket and want to take a gamble that the bank has no interest in pursuing their security interest in the property, have at it.
With all due respect to you and your knowledge, you don't know what you are talking about.
Once a tax sale goes to sale it is all over.