I'm not a real estate professional, or a lawyer, or any kind of expert on anything for that matter, but I have attended tax sales with a friend several times and have talked to tax assessor/collectors on occasion and might have a couple of ideas for you to check into.
If the property appears to be abandoned, go to your local tax office (be sure to check county and city if they are not collected by the same office). Ask them if it is in arrears and whether or not any liens are currently on it. You can usually do a check at the courthouse for title information including whether there is a mortgage on it, current ownership, etc. If you need help, usually a clerk will help you if you let her know you're researching because it is next door to you. There is a downside of that, and that is that there are some people out there that if they know someone is interested in a piece of property, they will try to obtain title to it and then make a profit by selling it to you. In our county, showing interest in a property immediately sets off a red flag to investors.
If the only thing owed on it is taxes, especially back taxes, you might be able to get the current owners to "quit claim" it to you for a nominal fee. That would allow them to rid the tax liability and transfer that to you. Also check for any liens for mowing, whether it has been condemned,etc. Most taxing authorities are eager to get property back on the tax rolls and will not stand in your way.
Two other things I can think of off the top of my head would be making sure it isn't tied up in probate. It is still doable, but more complicated. My sister bought a condemned house that way, but it had about 15 heirs she had to deal with and get signatures from separately. She paid a lawyer to do that since she lived out of town. It was pricey, but worth it because there wasn't really anything majorly wrong with the house, but it had been vacant so long the the city condemned it when the neighbors complained. It's very possible you could handle all of this yourself. The other thing is whether you would be able to have a clear title for title insurance if you sell. One way around that is to keep both properties separate and sell them separately if you do decide to sell in the future.
Also, as far as demolition goes, you might be able to have someone come clear it for you in exchange for the materials. There is a growing movement of people who are looking for salvage material to build with or resale.
Hope that made some sense, like I said I'm not a professional. And it would totally depend on your state, area, etc.