Just a full check up. And if the vet wants to do other tests they will talk to you about it.
If you can afford it, I would request a 5 stage vetting. A general vetting is a must at the least.
It depends on what you want to do with it, but xrays are something to consider if you want to compete or do strenuous work. Blood work is also something to look at, in case any sedatives have been administrated before you view the horse.
I would also take it on trial if you can (or do a home trial where the horse stays on the yard it is on as many people don't do trials any other way), and be sure to test the things the owner claims it can do.
Unless you are used to buying/evaluating horses, I would definitely take someone more knowledgeable with you.
If you are buying from a dealer or someone you don't know, there are a few "dodgy dealer and seller" pages on Facebook. It's always worth doing a quick search of them to see if their name has popped up.
Get the horse's passport and make sure it's the right one.
Be careful. Whilst majority of sellers are genuine, there are some out there that have no issues selling on a problem horse and marking it as safe, or selling a damaged horse without disclosing it, or even selling a 30 year old and saying it's 12.