internal parts through the oil filler neck

If it's possible, look at the internal parts through the oil filler neck (Don't open the oil cap while the engine is running!)

Dirty engine Clean engine

Oil sludge inside the engine

With the engine turned off, try to open the engine oil cap and look inside through the oil filler hole. Use a small flashlight. If it's possible to see any internal parts there, check their condition. What you are looking for is the oil sludge - this thick black buildup on internal parts you can see on the lower photo (this is how the sludged up engine looks inside, with the valve cover removed). If you see a lot of black oil sludge buildup inside the engine it means that either the engine oil hasn't been changed for a long time, or the engine has been overheated, or poor quality oil was used.
In either case, avoid buying such a car - sooner or later it will have serious engine problems.
Well maintained engine usually looks pretty clean inside; if you check few cars of the same model it's easy to see the difference. However, again, if the engine looks clean inside it doesn't mean that it has no other possible problems - as a last step, have the vehicle inspected by a mechanic of your choice.

If you find evidence of badly made repairs, avoid any such car.

If you find evidence that the engine has been repaired with inaccuracy (too much bulging of gasket maker, as in the image, loose bolts, stripped treads, altered wiring, broken fastenings, missing bolts or covers, etc.), avoid buying such a car because quality repair is possible only by very skilled mechanics. Just imagine how the engine has been repaired on the inside if it looks this way on the outside?

Timing belt

Many cars, especially with four-cylinder engine, have a timing belt that needs to be replaced at a certain interval - usually between 60,000 and 100,000 miles (100,000 and 160,000 km). If it wasn't replaced by the previous owner, you will have to do that. It's difficult to check it on the car because the timing belt is protected by the protective covers. The only way is to ask the previous owner if they have a receipt. Sometimes though, dealers place the timing belt replacement sticker somewhere on the top of the engine (see the picture) that indicates the date and the mileage when the belt was replaced.

Check the smoke

Blue smoke at start-up may indicate engine problems - avoid such a car. Black smoke means the engine consumes too much gas - possible problem with fuel injection.
Normally, there should be no smoke at all (Diesel engine may have slight black smoke at a cold start - it's normal). A small amount of white water steam and water condensate dripping from the exhaust is normal.

Listen for noises when the engine is running

There shouldn't be strong noises, coming from the engine under any condition with no matter is the engine cold or hot. By the way, knocking or tapping at a cold start is one of the indicator of poor maintenance. Knocking, tapping or rattling noises indicate excessive wear of internal engine parts. Whistling may be caused by loose belt. If the engine makes too much noises, avoid buying such a used car (however, Diesel engines are always more noisy, it's normal).

Test drive

Try to test drive a car for as long as you can. Try to accelerate, decelerate, take it on the highway if it's possible. The more you drive, the more chances you discover possible problems. If it's your first car, try to test drive few different cars of the same model to have better idea. If you feel anything that may indicate possible engine problem (e.g vibrations, stumbling, misfiring, delay during acceleration, rattling noise, smoke, rough idling, etc.) avoid buying such a car. If you have any hesitation about the way the car drives, shop around, there are plenty of used cars available. Sometimes a sales person may try to push you to buy a car today and now because "tomorrow I won't give you this price" or "I have the customer who will buy this car tomorrow" - Don't rush, take your time to think, the more you shop around, the more chances to find the right car for the right price.