Buying a car, this is assuming it is a private party. Some of the things are a crossover if buying from a dealer but from a dealer there are usually less to check.
Before you even start the engine. Look under the vehicle, do you see anything dripping? Water, tranny fluid, oil etc? Look at the tires. Are they wearing evenly? Is one side wearing more than the other? Check all the windows to see if they all go up and down, and also listen for squeaks or gears binding. When the windows are up hear any wind whistling through it? When the windows are down check the condition of the seals (the rubber) around the glass. Walk the body with a magnet, if it stops sticking, it is a good chance they had body work done (there is Bondo under the paint). Check the corners of the body and under the corner of the trunk, see if there is any rust there. Also see if there is any rumples under the paint, could be rust also. Check to see if the paint on the body matches that in the doorjambs, and inside the trunk and under the hood. If different, maybe it was repainted after a collision. Check all the lights, turn signals, etc. Make sure there is no short. Check all the keys; make sure they work. Check the wipers.
Then check under the hood. See what look like rusted water sprayed on the under side of the hood? Could have an overheating problem. Are the hoses brittle, frayed or bulged? Is the wiring harness loaded with electrician’s tape repairs? Is there a lot of oil leaking from the valvecovers? Any moisture around the freeze plugs? Check the tranny fluid, any metal shaving in it? Burnt smell? Does it have a dark molasses color? Check the oil, does it look like café au le, or a milkshake? See any beads of water? If you can grab the fan blades and give them a shake. Does it wiggle? Check the radiator if you can to see if you see any leak.
Now you start it up. Does it start quick or do you have to lean on the ignition to get it to fire? Once it starts do the idiot lights on the dash go out or is any still on? While it is idling, is it a smooth purr, or is it loping? Do you hear any knocks up top, or down deep? Could be a bad cam, valves, valves hitting a piston top, or worse, it will need new rods. Is the exhaust clear? Do you see anything when you stomp the gas? When you let off the gas does it return to the idle you started with?
On the road, take your hands off the wheel (keep them real close) does the vehicle run straight, or does it pull to one side? Does it want to pull when you brake? Does it go into passing gear easy without hesitation? Is there excessive wobbling up front at speeds over 45 mph? Listen to those windows again, any wind whipping through them? Does the heater get warm? Is the A/C cold?
If it passes those things, the major ones anyhow, I would take a chance if the price was a really good deal.