You are silent as to where you bought the property. Eviction and Lease laws are STATE SPECIFIC. Even worse some local municipalities are involved in the process to the point those cities are not investing in. So even if you were in Illinois anything we do in the Southern two thirds of the state just might not work in Chicago and the Collar Counties. So any advice you get here should be used in the areas where the poster is from with the understanding that they might not work in your area.
That being said here is what I think: Your current residents drove their family member to sell the building because they were happy to take advantage of him or her.
1. Pick your fights on a first come first served basis. Send a letter to each resident asking if they have a signed lease in their possession, and could they provide a copy to your office. Otherwise, you must assume they are renting on a month-to-month basis and subject to those terms and conditions as defined by state and local law. Don't explain the terms and conditions. They will be getting enough free legal advice as it is.
2. Contact the local landlord's association and get the names of three good eviction attorneys.
3. After choosing and speaking to an attorney use the correct way to raise rents and do it immediately. Make it a "modest increase" so as not to chase them all out at the same time.
4. Evict one at a time. If the other three are paying let them stay at the new rent rate. A 25% vacancy rate is easier on vandalism, other nonsense and your checkbook than 100% vacancy. You can always raise the rent again with proper notice.
5. Don't run out and sign new leases, just get new applications. Keep all those who remain on a month-to-month basis. With proper notice this makes eviction without cause an option.
You may need to evict every single one of them. The reason for this is because their prior landlord basically ruined them as they destroyed him. If they hadn't destroyed him he would still own the building. Rehabilitating your residents is hard.
It only takes one bad resident to chase out all of your good residents. Get a mentor at the local association and don't be afraid to buy them breakfast, lunch or dinner on their schedule. You should also get established with a quality screening company.
Good Luck and Good Investing