Hi Shiva - Congratulations on your new purchase!! When we purchased our triplex, it was fully rented. The way we approached it was to review the existing contracts and make sure that we had the deposits in place - the ones who didn't have deposits we worked out a monthly payment till it was fully funded. We rewrote a new contract since it was month to month - We had to evict one tenant however because it was an owner occupied situation, so we had to take possession within 60 days.
In terms of approaching them, I just knocked on the door and introduced myself. I let them know that we were the new owners and that we would be moving in shortly. The first order of business was to find if there was any issues or deferred maintenance that needed to be addressed. There were couple of small issues that had gone unrepaired and we were happy to get it fixed up as no one want to see their own property in disrepair, plus it gave us instant rapport with the tenant, who knew were were willing to take a hands on approach to keep the place up. We went through a check list to confirm everything else was in order.
If you plan to keep your tenants, introduce yourself to them and let them know that you are the new owner and find out what issues they've been dealing with. This will give you a good heads up on the type of people they are and if you approach it from the standpoint of providing a positive customer experience, it should be an informative and productive meeting.
Of course this is all assuming you plan to keep your tenants. Just be careful about listening to seller about the type of tenants they are - the seller I bought from told me about and endorsed the "best tenants" and it turned out that they were slow to pay and were quick to leave me high and dry with no 30 day notice. The "problem tenant" he warned me about turned out to be one the best tenants I ever had and I was sad to see them leave when the kids outgrew the apartment.
Owning rental properties is as much a people business as a cash flow business and like any business you want to have happy repeat customers. Make sure your contracts and deposits are in place and that your tenant knows how to get a hold of you and where to make their payments. Be sure to provide and have them sign a copy of the rules and regulations - they don't seem very important until they start breaking rules you didn't even think you needed...that will be a fun one...if you plan to increase rents, do it when they are most likely to expect it like as soon as you take over or one year since their last rent increase/ contract. I would strongly suggest that you get a copy of your local laws regarding landlords rights and responsibilities. This can save you a lot of frustration if you get into any sticky situations where your not sure what you can do legally - trust me - your tenant will be experts on these rules ( or at least what they learned on people's court)
I'm sure you will get a lot more suggestions from the seasoned pros here who have seen it all....hope this was helpful. I look forward to reading your updates. Good luck on your new venture!!