I have an '05 Avalanche and maintain a fleet that includes Tahoe and Silverado in from 2000-2005. The 4L60E transmissions have been rebuilt on three before 150k miles. Weak sun shell and planetary pilot bearings mostly. Later models are doing better. Check the fluid and service it according to the owners manual recommendations.
'04 had some problems with sub standard wiring. Fewer strands and less flexible. Breaks in the wiring created some "reduced engine power" drive defaults due to communications errors between the throttle body and pedal.
The 5.3L engine is a good one. I have had two with the intake manifold gasket leaks. Reving to high idle when cold and erratic idle are signs.
Waterpumps seem to go around 120k-150k miles. look for seeps dripping on to the crank pulley, or stains running back along the oil pan. Plan on rebuilding the starter and alternator about the same time. Weak cranking , or low voltage might be signs. Pretty easy fixes.
6.0L has tons of power, but the mileage and oil consumption leaves lots to be desired. My preference goes to the 5.3L.
Side gears on the open differentials were soft and flaked. An upgraded set was offered. Those with the G80 differentials did not have this problem. Make that a must find item. The Z71 should have it, just check to make sure. GT5 4.10 gearing pulls great , but you'll need taller tires to get decent highway mileage. I like mine , but higher geared rigs are getting better mileage than I. Check for slack in the differential. Place it in reverse, the to drive with the brake on an listen for a clunk or excessive slack. U-joints might be an issue for those who have been swimming frequently.
Suspension parts wear. Check the ball joints , idler, pitman, etc. They really do hold up pretty well, but after 150k it might be time. Look for tire wear and slack in the steering. Some feathering on IFS front suspension can be normal. Shocks can soften that if they are in good condition. The Z71 usually had pretty good upgrade shocks.
Instrument cluster failures are common. Stepper motors mostly. If you are handy, you can fix those yourself. Otherwise plan on $300 for a reman and flash for the install.
The interior switch illumination is a problem. Those little bulbs burn out. Plan on at least $20 for each steering wheel switch with a burned out bulb. You can tear them apart and fix them if you're handy. That ends up about $2 each. The radio has the same bulbs. It is more difficult, but can be fixed too. The CD player on the stock radio can be an issue. Many have had the CD player gobble a disk and not give it back. There is a small plastic part that breaks. Not replaceable. A CD mechanism from a used radio can be swapped in , or a replacement radio installed. They are theftlocked to the VIN so dealer programming is required.
Battery acid can eat through the plastic line for the washer pump next to the battery. If the washer fluid is low or leaks, it is likely that line. Easily fixed with a sealed battery and some hose.
Every rig has its weak points. No deal breakers for this one in my book. The Silverado that I drive has 204k miles on it and still a trusty steed. I trust my Avalanche too at less than half that mileage.
You may find some useful information here:
Z71 Tahoe/Suburban.com Forum
My rig here:
BLT Offroad Avalanche