The first thing you would do in this situation is explain to your tenant what is happening. Let them know that you want them to move out and if they do not, you will evict them. Educate them on how an eviction can harm them and explain that it will likely be expensive for both of you. This is why you will pay them a certain amount to move out now. This amount can be whatever you decide, however it is often suggested that somewhere around $400 or $500 will do the trick.
Next, explain that you want the apartment cleaned and any repairs done at least have them pay you for them by the time they are ready to leave. Let them know when you want them to vacate the property. Make sure that you give them enough time to get everything out, but not too much so that they could do more damage. Often thirty days is given, much like in any other process. Make sure that you have a written notice that you have kept a copy of. Once they have moved their things out and cleaned, go in to inspect the property with them to make sure it is in the condition you asked just as you would with any other tenant. Take pictures of how it looks. Now all you need to do is have them sign paperwork that states that they are officially moved out and you can hand over whatever amount of money you agreed to pay them.
While it may seem strange to pay someone to leave, it often does work out in your favor as they pay for repairs and such. Let us know what you think of this process and whether you would ever use it.