I've been meaning to make a formal list, so this is a great question. It's all part of your due diligence period. I'm sure if you check for a due diligence check list you can find something better, but here's what I look for:
-Get an inspector to get you a full inspection. Make sure everything is up to code and has permits. If you plan on doing any expansions or repairs make sure they will be permitted as well
Ask the city for any and all information they have about the property, it's free and should take ten minutes.
Have the title company run a title search for you and get title insurance so nothing comes back to bite you later.
Get your contractor to walk the property to look for anything your inspector might have missed and get a quote for any repairs you may need.
Talk to a property manager to make sure it will rent for what you need it to. Ask if there's anything you should be aware of about the neighborhood that will impact your rent prices. ie. new construction, gentrification projects, a new Amazon plant opening, etc.
you MUST get the rent rolls and a 12 month rental history and P&L. Make sure that even in the worst month you would still be positive cash flow. Find out your rental demographics. Are you going to inherit any section 8 tenants? You'll definitely want to know that information. Is there an HOA? They take a lot of your cashflow and a lot of control you have over your property, buyer beware.
Have multiple exit strategies if you need them and make sure you are conservative with all your numbers so you don't get caught with your pants down. I would say at a minimum, have your Realtor, PM, and another investor look at the numbers. Make sure the cap rate is realistic for the area you are in. ie. If you found an 8.5 cap in a A neighborhood, it's probably too good to be true. So just be aware that the numbers can be explained and make sense.
I'm sure I'm forgetting something but I hope this was helpful.