First off, you definitely have to find out your state rules. If there leases in place, those must be honored. Even the month to month ones.
In our situation, we bought a 13 unit building that was in distress and 20% under market on rents. But since the condition of the units were in such bad shape, I honestly couldn't in good conscience raise the rents to market rent right away.
Upon take-over, two tenants abandoned ship. We started by rehabbing those 2 empty units. When our current residents saw how nice these new units were, they willingly paid the $115 rent increase to get a nicer apartment. These were long time residents and very good renters. This pattern continued in the good tenants wanted to stay and move into the new units and the others left, either by eviction or on their own accord.
The whole process took us a year. We usually had two empty units at a time that we were rehabbing. And by doing this unit by unit, we had positive cash flow on the deal at all times. It took planning, diligence and perseverance, but when I start this process again, I plan on trying the same plan of action. Good luck and keep us updated!