BP has a list of tenant screening companies if you don't already have a resource.  

You still need to screen the parent's and the students. 

There is an aluminum screen door,  then the broken door in photo leading to a 3-seasons room type enclosed poarch. 

Shards of broken glass every where.

My plans are to replace elevator, renovate apt to higher standards, and start screening for top quality clients.

I would recommend reading the guide to tenant screening that they have here on BP. 

I agree with Bryan Neal....I do my own screening and very rarely contact the previous landlord in case they have not given notice. 

Here is a quick list of things to look for when DFD:•Missing Window Blinds•Dead / Overgrown Lawn•Bad Roof•Lots of Trash and Cars with Filthy Windows–No License Plates, Flat Tires•Broken Windows•Lots of Deferred Maintenance / General Disrepair•Peeling Paint•Overgrown Trees–Lots of Tree Litter on Roof•Fire Damage•Graffiti•Owner Board Up•Notices Posted on Doors / in Windows•Uncollected Mail / Newspapers / Phone Books in Driveway/Mailbox•General AbandonmentWrite down the address & take notes, go home and research owner info/mailing address using local title company, send letter.

If you acquire a 3-4 unit property and there are tenants there already, what is the best legal method you have done to remove them so you can pick your own tenants base on your screening methods. 

Broken screens on windows, non maintained back yards, mail pilled up.