Couple things I would look for:
1.) Check the hubs on the front and rear of the truck ($175 + labor a piece to fix). Jack the tire up and then shake the top and bottom. Pushing the wheel towards the motor and away from the motor. If it moves, you need new hubs. I wouldn't want to see 1/2" of play or more.
2.) Check if there's oil in the coolant. It will look like brown baby poo in the coolant. That's a symptom of a shot waterpump, oil cooler, or something more serious (heads/head gasket, etc).
3.) Check if there's smoke coming out of the motor. At the same time, you can check a number of other parts to the truck, by making the guys start the truck cold. Get there early in the AM, tell him 10am, and then show up at 9am. You want that motor ICE cold before you start it. Check with your hand on the temperature gauge when you prime it. You'll want to prime it, see that the glow plugs work and that it starts as it should. You don't want a hard starting truck, unless you're ready for that. You'll want to watch the smoke come out on start up. White (blue) smoke means oil is leaking in to the cylinder.
4.) Hop up on the front of the truck and peer into the valley behind the fuel bowl. Under the power stroke logo plastic piece is the fuel bowl, and behind that you'll see fuel lines and then the crevice above the engine is where oil will pool. You don't want oil or fuel sitting in there. So while the motor is cold. Stick your hand back there and see if there's oil or fuel.
5.) When test driving. See that the turbo spools correctly and that it shifts through the gears correctly (no jumps, or hard shifts). It helps to bring someone who has a good condition 7.3 with you. Check behind the turbo by the exhaust for soot. Check on the tubes leading into and out of the turbo for soot. You don't want diesel soot in the engine bay.
6.) To test the 4x4, you can either a) put your face to the curb and look at the axle while he drives back and forth b) put the truck in 4x4, and then turn it all the way to the right or left, you should feel a shudder.
7.) Obviously, take it to see if there's any codes being thrown at an AutoZone or an Advance Auto. They'll do it for free. While down there, check for electrical burns, and check the PCM for any electrical burn.
8.) Bring someone with who can run through an injector buzz test and an inject contribution test. You'll want to know that there are no shot injectors, PCM, IDM, etc. Injectors shouldn't be contributing at different levels, they should all be uniform. You want no loss of compression. Could be a cracked head, bad head gasket, or even a faulty injector ($200 a piece + labor)
9.) Check the interior thoroughly. Remember if you beat up your interior, you probably beat up your truck. Good interiors are usually a good indicator that someone loves their truck (not always true, though)
10.) Check the frame for rust and for weld marks. Weld marks mean the chassis was bent and has been straightened out. You will never get a chassis perfectly straight. It will always be slightly off, if it's been in an accident. Which could mean drivetrain issues down the road or a lot of tires down the road.
11.) When driving, take your hands of the wheel at about 45 mph and see if the truck pulls in a certain direction. Could be a sign of crabbing, worn hub, very poor alignment, etc. Then perform the same test while placing your foot on the brake. If the truck pulls to a certain side you may have a faulty brake (remanned brakes should run you a $105 + labor a wheel).
12.) If the truck is over 135,000 miles, that tranny should be rebuilt or replaced. If it has been rebuilt or replaced, get the receipt (people will save that receipt). If they have the receipt, call the shop and confirm. If they don't, call the shop and check their records. Rebuilt trannies are pretty much a necessity in these trucks. The fewer the miles on it the better. I got mine with 10,000 miles on the tranny, all forged gears and billet torque converter.
That's just some things that I would check on my own. If you can get through that list with everything being gravy, then I would take it to a mechanic, spend the $100 and have them diagnose any problems with the truck. $100 could save you $3500 in repairs by making you not buy the truck. Take off the cost to repair from your offer, and you should be good to go. Always remember to inspect, inspect, inspect. And don't lose sight of your target price. If you have $15,000 to pick one up, you better get the truck for $15,000 less taxes/etc and less repair costs. I spent a little more fixing mine up than I wanted to, but she's my diamond in the rough. I also told the guy I was going to have to spend $3500 to repaint the truck, so his asking price of $13,900, just became a best offer of $10,400.
Oh, and aftermarket anything doesn't drive the price up. Whoop-di-do you have big tires and wheels. Tell them you don't like them and you're going to replace them, if they say that the tires and wheels alone make the truck worth $2,300 more. Aftermarket anything doesn't improve the price. In addition, if they have a short bed, tell them you're looking for a long bed, and you would offer $500 less because of that difference. If they have a long bed, you want a short bed. Don't waste your money on anything without 4x4.