If I were inspecting a machine, it would go something like this.
Basic external look at the machine - hours, appearance, any visible leaks, tires, bodywork dents and bends, completeness (3pt, attachements, handles, lights etc.) and overall look at how well loved or abused the machine has been. Well greased and lubed indicates more love in my book. Under 1000 hours - it's been lightly used - 3000+ would be well used.
Remove the covers and do the same for the engine compartment. Look for leaks, any indication of major surgery, engine removal or replacement etc. Pull the dipstick and look at the cleanliness of the oil. Very clean - can't learn anything as it was just changed - pretty clean - likely well maintained. If there is an hours indicator on the filter, or a sticker or other indicating last / next oil change - the owner cared. Check the air filter restriction indicator - does it appear red? Check the rad fluid etc.
Start the engine up and listen - anything unusual? Then I'd start with the hydraulics - everything move properly work properly - reliefs working. Then try the 3pt and do the same.
Then I'd try the drive - forward and back, 2 whl and 4 whl if applicable, straight ahead and turning. I'd be feeling and listening for any strange noises, stiffness, clunks etc.
Finally check the electrics - wires, switches, lights etc., all function properly. Is the hour meter turning?
Of course if anything comes up, I'd research the cost to repair that.
Now a good dealer will do all this for you for a nominal fee. Before I bought my 855 privately, I took it to the local dealer (who gets all service business going forward but had not really dealt with me prior) and their head mechaninc spent 30 mins with me going over the tractor similar to above, highlighted a couple of minor things to note and one major thing - then gave me a rough estimate to fix it - all for free. Made my negotiation with the vendor simple.
BigEddy