I had a '97 K2500 with the 5.7 towing a 5000 lb trailer. It did OK, but going uphill was a challenge. Not the fastest by any means. My thought was "My next truck will have a bigger engine!".
Fast forward 2 trucks - I now have a 2000 K3500 with the 7.4. Even towing 6000 lbs, it still isn't the fastest up hill.
The other weekend I brought a 8k trailer back for a friend with my 7.4. I did it, but I had to watch the hills and downshift to second.
I could not imagine towing 10k with a 5.7.
Looking at the towing guide: http://www.trailerlife.com/wp-content/uploads/2012/01/Trailer-Life-Towing-Guide-2000.pdf
C3500 (2WD) with 5.7 must have 4.56 gears to tow 10k. If you have the 4.10 gears, you only get 8600 lbs.
Also, you didn't mention 2WD or 4WD - if 4WD, you only get to tow 8300 lbs.
