Hey Edouard,
First of all, congrats on your first investment property! That's awesome!
I manage property for a living so I've been in this dilemma many times. The problem is there's no one right answer or solution that works every time. You can read about it and get all the advice you want but when the rubber meets the road, that all goes out the window. My one piece of advice is that when dealing with people, you need to approach the situation with empathy and treat them with respect. Anytime I'm serving a 5/10/30 day notice or having an awkward conversation like this, approach them like a friend and turn up the heat if the situation calls for it. In this situation, I'd reach out to them and tell them something along the lines of "Hello Mr. Smith, my name is Edouard. As you probably know I purchased the building a few days ago and I'd like to set up a time to stop by and meet you." If we take over a property in Chicago with security deposits, we always return them to minimize liability. If this is the case, and you let them know that you have a check for $X, they will always taking the meeting. When you meet them in person you can feel out the situation and gauge where to go next. Who knows....they may have a legitimate reason that they weren't paying rent and want to pay going forward (if this is the case, I would still go through standard procedures of qualifying/screening them). If they don't want to stay or you get a bad vibe and aren't wanting to keep them as tenants, I would serve them a 30 day notice (have this filled out beforehand just in case) and explain that they need to be out at the end of the following month. I'm always as nice as possible and try to work with the tenant as much as possible (sometimes offering to find them a new place or offering to pay for a moving truck) in order to prevent an eviction. If you haven't been through an eviction, it is very time consuming, exhausting, and costly....especially in Chicago!!! If they don't seem willing to voluntarily leave, I would go the cash for keys route. I've seen people move out for as little as $500 but it just depends on how desperate the tenant is. If they aren't biting on that then you can drop the hammer and serve them with the 5-day. I always walk them through how the process works and remind them in a nice way that we'll put a judgement on their record (ruining their credit), garnish their wages, and ruin they're tenant history (with an eviction on their record). I reassure them that that's worst case scenario and we definitely want to push for an amicable solution if possible. If they are career tenants working the system, they won't be phased. If they're decent people, they'll wise up and work with you instead of against you.
This is the method that I've used dozen (if not hundreds) of times with great success. Sorry for the long winded response....hope it helps! If you need anything, don't hesitate to PM me.