$4000-$5000 will buy a nice, updated flat screen. To make sure you get something compatible with upcoming technology, make sure the unit has HDMI inputs - two if possible, but one for sure. Try to get a unit that displays true 1080p - there's not many out there right now. The new DVD format Blu-ray will be capable of 1080p contect, as will the PS3 gaming system. There aren't that many sets out now with 1080p, so you may need to wait for the market to catch up a bit. If you're anxious to get a set now, and can't wait for true 1080p, then go for current pq - the Pioneer Elite at 1080i.
You need to decide what size screen your room can handle. You may opt for a projector and a large screen if the wall is large enough, and room dark enough. Flat screens like LCD's and plasma's are great for pq and viewing angles, but sacrifice size. Pioneer Elite probably makes some of the best plasma displays, so if you're leaning towards plasma, research Elite. LCD isn't competitive in that size screen...yet. For 37" and smaller, go LCD. 42" are becoming more popular, but the largest LCD sets are out of your budget. The screen life of an LCD is 60,000+ hours, and there are no burn in issues. The color depth is said to be a bit behind plasma, as well as black levels lacking slightly. You need to compare two side by side to see the difference. Of course, a very good LCD is going to look better than a cheaply made plasma screen.
High definition signals would be 720p/1080i or 1080p. Your screen resolution would need to be rated at minimum those parameters. Those are lines of resolution. The highest HDTV resolution is 1920 x 1080 pixels. I'm not sure how versed you are with i/p, but I'll assume you know what they mean. Just check the frame rate - 60 frames per second is the highest available.
A unit with a built in ATSC tuner is nice if you can receive over the air high definition programming from local stations. ABC, NBC, CBS, and PBS offer free over the air signals. If you live in a metro area, you may get by with an inexpensive rabbit ear type antenna inside. If you're a distance away, you may need a larger antenna in the attic or roof.
Check this website for stations in your area. My parents get about 7 HD channels and have no need to purchase an expensive HD box since they only watch the national broadcasting stations.
www.antennaweb.org