I asked our current Treasurer as she deals with stuff like this quite often:. Here is her response.

• If the current owner has a family member that would probate the estate and is made executor of the estate and can deed property over to the City with the City paying the back taxes is the cleanest option. 
• If that isn't going to happen, the City can go and purchase the tax certificate from Custer County as they sell the tax cert to third parties.  The City would then have to go through the tax deed process to get a treasurer's deed to the property.  
• Last option is to wait until the County takes it for tax deed and ask that they surplus the property to the City.