Well, it's a bit too late because you already bought the car, but my check list when I go to see a vehicle for sale is something like this:
• When engine cold:
- check air filter (if it's too dirty - most of the time it means the rest of the car is neglected)
- look for any oil and coolant over the engine and transmission
- check the engine oil level (it should be b/w min. and max. on the oil dipstick)
- check the coolant reservoir level it should be at max. or b/w min and max., but no lower than min.
- check for oil and coolant leaks (under the engine and transmission)
- open engine oil cap and smell (does it smell like burned, wipe the cap with paper or a finger and see if leaves any deposits/sludge)
• Start the engine:
- listen of any noises
- look for any smoke
- take the car for a ride, look how quickly reaches working temperature (it shouldn't take more than 5-10 min.)
- listen for engine and transmission noises, especially when starting after being stopped and accelerating
- check if the car pools left or right
- listen for any noises and vibration when slowing down and stopping
• When engine is hot - driven at least 15-20 min.:
- open the hood and check if you smell any oil or coolant (if it smells either one - it means there is a leak)
- if automatic - check the transmission oil level (it should be b/w min. and max. on the oil dipstick), but no lower than min.
- look under the engine and transmission for any leaks
- look at the coolant reservoir - it should be over max. or above the middle b/w min. and max.
- make someone push the throttle while you're listening for any noises
If the car looks messy and dirty on the inside and out, most likely it means mechanically was also neglected.