Purchasing duplex we currently live in, with existing tenant

[Baley Whary]

Hello, we've been living in our duplex for 5 years and our landlord decided to sell and offered it to us first. We're going to be "accidental landlords" so to speak, and have current tenants in the unit next door. We've been up front about attempting to buy the house and they seem positive about it and want to stay. Currently, they're paying month to month at a lower than market rent rate. What's the best way to come in to this purchase regarding their (lack of) lease? Is it ok to sign a lease with them without running a background check (we really don't want to lose them right off the bat) and how much more can we ask them to pay in rent? The market value appraisal indicated that their share of rent should be $925, but the current landlord was only charging $750, and let them live next door for $700/month because the space would otherwise have been empty. We're also trying to get new windows installed immediately, which should significantly decrease their electric bill. How should we approach a lease signing with them? We have lived adjacent to them for a few years (it's a long story, but they were sharing our half of the duplex which was modified to two living spaces and they were downstairs), and we're friendly enough with them and they have history of paying rent, so we definitely want to keep them happy. I also want to at least raise their rent to $750, with the understanding that we will probably raise it another $50 after a year. Any help would be more than welcome! (Current status of sale: 95% likelihood it's going to happen in the next 2 months.)