@Dylan B.
If you buy a property you generally have to honor the existing lease until its termination date. From there you can go month to month or provide a 30 day notice prior to the end of the lease and have them move out when the lease expires. Leases don't expire just because the ownership changes hands.
Deposits SHOULD be transferred to you but sometimes they don't. If they don't you are still liable for them so know what they are. The downside is you won't know what the condition was when they moved in. So if you haven't done so yet, once you close schedule a time for a full walk though of each unit to document their condition and you can take any abnormal wear and tear or damage off from that point.
I would request copies of all the leases so you know when they terminate, the deposits, and other key facts of the lease. If the landlord has any background files or screening paperwork I'd request that too.
If tenants don't meet your requirements and their lease has expired I would look issue them notice (in AZ it's 30 days) that they need to move out. Be kind and upfront with them and hopefully you won't need to evict them.
Best of luck!