@Annie Ruffino Congrats on the contract! Those are hard to come by these days, so just getting it under contract is an accomplishment. I do property management and I always keep a tenant when I can. First the seller more than likely is not going to give them notice unless they are a bad tenant anyway. At this point they don't know if you are really going to get to closing. They are not going to get rid of a good tenant and then have a vacancy if you do not close. If you want them out you need to read their lease. If they are month to month they are probably going off of their original lease which normally says you have to give them 30 day notice to vacate. If no lease check out state laws for giving notice.
Do they not have a deposit with current seller? I would ask the seller if they did a background check. If they did then ask them to supply that. That way the tenant doesn't even know. I know that some people just let anyone live in their properties, but I would think that most do background checks, or at least call their employer, ask for paystubs, and/or check out previous residence. Ask for that information from seller. Ask the seller for rent roll to show if/when they have been paying. Honestly, with a tenant I inherit I really don't care about their past, I care about their present. I assume you walked the property? What was the condition? Were they taking care of the home? Was it clean or disgusting? If they are taking care of it and paying, I would keep them. I have also found that all the background checks in the world do not detect dirty or crazy! I may think I am putting a better tenant in there and turns out they are crazy!
As far as raising rent. If you are bringing it to market rents and they can not go anywhere else for cheaper and have to pay to move plus the headache to do so, I have found they had rather stay and increase their rent. I have one right now that has been a challenge. The new buyer bought it with a hard money lone (I sold it to my investor). The current tenants were paying between $650 - $850. Market rents are $1300. The new owner wanted to give them notice because he has the money now for reno with the hard money loan and later he will not. I gave them notice and they freaked out. They did not want to move. Even though their rents were low they paid, and on time. They were willing to stay and raise their rent to $1200 per month! This was not good enough for the buyer he still wanted to rehab. So now they have agreed to stay for $1300 per month and let him rehab while they are in there!!! That is going to be difficult but is the perfect scenario. Now we do not have to worry about vacancy, adverting, all the time and money it takes to get new tenants. Some of them doubled their rent and were fine with it. Honestly, I did not even ask if they had background checks. I don't care. As long as their units were cared for, somewhat clean and they pay on time.... that is all I can ask for. I have many tenants that we thoroughly vetted. They had good jobs, they always paid current landlord on time, no issues, etc... then they moved into our unit and lost their job, didn't pay, had animals they didn't tell us about, constantly complained about repairs that cost a lot of money when they were not paying, etc... There was no way to foresee that coming. On paper they were good. So I had much rather keep a good current tenant that has proof of being good then to speculate on a new one.
Sorry I know that was a bit of rambling! Hope it helped at all. Happy investing!