Big thing to look for is proper maintainence from previous owner(s?), as the maintainence is the major cost ona 1k car...if you buy it for 1k, but put 3k into it in the first year...shoulda just bought a 5k car and saved yourself the trouble now, and down the road..
See, I have a different view on this. If you put 3k into a car, you're going to have a lot of new parts on it and a mechanic who has gotten to know it somewhat. Depending on your situation and what kind of cars we're talking about (obviously not all cars at a given price point are comparable), you may end up coming out ahead doing this.
If you buy a 5k car, great, you've got a car that SEEMS nice. But it can be really hard to tell what's going to go wrong, and if you stretched to afford it, you've got no backup. You've got to wonder about the timing belt, the heater core, the wiper fluid pump, the transmission (especially automatic), and a bunch of other things that might go wrong that most people wouldn't think of and/or are hard to know for certain anyway. Some of those questions will have been dealt with after you put 3k of repairs into a car.
Me, I'd rather have a 1k car and the ability to put 3k into fixing it than a 5k car and no reserve if it blows a head gasket the day after I buy it (which DOES happen- remember that used cars are for sale for a reason). I don't care about the resale value of my cars, because I drive my cars into the ground. I care about the value of the money I spend on my car situation...and in my situation, I can get more out of buying a beater and being able to fix it if I have to. I guess what I'm saying is, buying a 5k car instead of a 1k car is not what I would generally consider a reliable way to avoid trouble. Put another way, I'd like to choose where the 3k worth of repairs are going to go rather than trusting the previous owner to have done it for me.
Remember that we're probably talking to a poor college kid who wants to get a reliable car for cheap, not a gearhead with full access to his daddy's racing garage and a bunch of cousins who will help him do the valves over the weekend, so that's where my advice is pointed.
As for finding the best car, there's a lot that could be said. Look for signs of repair or misuse- moldy smell, heavy air freshener, aftermarket seatcovers, heavily worn or brand new pedal covers, rust in the trunk, freshly cleaned engine, doors don't close smoothly. If it's an AT, count the gears as you drive it and make sure it has as many as it should. Look at the tires- is the tread wear balanced? Check all the CV joints visually and make sure to turn tightly both directions under throttle to see if they clunk (if it has power steering, don't hold the wheel against the stops, or you can damage the pump). Look for engine oil under the air filter. Make sure the AC, heat, lights, windows, and wipers work, or you can definitely fix them, or you can live without them. Smell the exhaust. Drive on a bumpy surface. Look under the seats. Check if it has a spare, jack, and tools.
The best way to buy a good car is to keep from buying a bad car. Be ready to take your time if you can, and be ready to walk away. In fact, expect to walk away. I spent a month looking for my last car, including a 120 mile trip to the nearest big city. Eventually I found my 1993 Toyota Tercel with 77k for about $2000 all told on eBay. Took the bus 600 miles, checked it out thoroughly, swiped my card, drove home, put it through hell, and it's still a reliable backup over 100k hard miles later with no work more serious than a couple of CV joints. Saved a lot of money and peace of mind. It's the base model- no tach, no passenger side mirror, no AT, no trip odometer, no intermittent wipers- so it didn't have much value in the marketplace, but it had a lot of value to me.