I am not too familiar with the body style pre-2007. However, I took a look at a car reliability website I am familiar with (and also a member of) called TrueDelta. It appears that the issue that stands out the most (for that particular year) is a fuel pump replacement at or just over 100,000 miles. Other issues, although not across the board, included A/C or heater problems, and some leaking issues reported. Of course, this does not mean that these are the only problems you will find, and you could very well get lucky and have no problems at all. It really just depends.
On a side note - I have a family member that purchased a 2002 Tahoe the year it came out and still owns and still uses it as their daily driver. It has just over 115,000, and they have had zero major issues with it (minus the expected tire changes, brake work, and I believe a new battery a while back). They love it, and don't even consider buying a newer one.
Feel free to take a look at the site yourself. The link can be found
HERE. Click on reliability in the menu bar across the top. On that page, in the drop down menu labeled "Make" choose "Chevrolet" and sign in as a guest. On the next page, in the drop down menu labeled "Chevrolet" choose "Tahoe/Suburban". Scroll down to find the "2001" "Tahoe / Suburban / Avalanche" link. Click on that and you will be taken to a new page where members of the site have posted problems with their vehicles, and the mileage it occurred on. Read through, and get a feel for some of the issues present.
I hope that helps! Let me know if there is anything else you need. Good luck with the purchase, and let us know how it goes!
---------- Post added at 02:17 PM ---------- Previous post was at 11:22 AM ----------
Of course, the other thing to do would be to check over the forum in this section. I am sure there are plenty of posts where members have asked about issues and/or problems they were having with their vehicles. You could probably even do a search specifically for 2001 or the year that you are looking for.