Can an abandoned property be claimed by paying back taxes owed on the property. Specifically - taxes have not been paid on this abandoned property since 2006 - this is 2014 - can I claim the property just by paying the back taxes?

If you are the original owner and the city agrees to it then it might happen. Other than that, the property would go through a lein sale for taxes due. The city would have ownership of the property.

No. If the local government decides to sell a property due to unpaid taxes, they will sell it at auction. You may buy it then. Just paying the back taxes is a gift to the owner of record.