I'm going to be doing a bit of heavy towing with my truck. Up to and sometimes over its rated weights (approx 7000lbs). What can I do to add the necessary power and transmission performance that I'm going to need?
tl dr - Upgrade to a 2500 or 2500HD pick-up.
There are several issues you'll notice when hauling at the maximum and over the maximum of the towing capacity of the Avalanche.
First off, the 2007 & 2008 MYs came with the 4L60e transmission. Why GM ever put these in a pickup truck is beside me. These were meant to go into passenger cars. While they are supposedly supposed to be able to handle 400 lb ft of torque, when you throw them with a huge load, they tend to fail rapidly. The only cure for this is to get a 3/4 ton pick-up with the 4L85e which can handle the heavy loads. This doesn't mean the 4L60e will expire on the first load, it just means it will fail a lot faster with consistent towing. Starting in 2009, the Avalanche came with the 6L80e, which could also handle the loads. The only alternative is to actually modify your truck with the a 4L85/90e or 6L80e transmission. GM finally decided the 4L60e wasn't going to cut it for their pick-ups (after a lot of customer complaints), so put the beefier transmissions in the rigs to support towing a lot better. Please note, if your Avalanche is a 4wd vehicle, this adds a whole other complexity to getting this done.
If the vehicle is going to be pulling any hills, the 5.3L engine will just not have enough torque. You will be lugging up the hills in 2nd gear driving at 35mph. This can be overcome by putting a larger engine (6.0L or newer 6.2L), which will have a better torque curve and plenty of power. There are also head/cam packages which could be added to your 5.3L, but it costs money either way. Another option is to put a supercharger on the stock 5.3L. As long as you don't go crazy, you can boost your engine without too much of an issue, but the life expectancy of the engine will go down, so be prepared. Don't forget you'll need to have the engine control module tuned for the specific setup as well.
You would also need to ensure the rear differential is at least a 4.11:1 ratio drive. If you have the 3.73:1 or (heaven forbid) 3.42:1 ratios, your in for some mighty arduous towing. Again, if this is a 4wd model, you'll have to do work on the front end as well as the back end for the gear change.
The last thing you need to worry about is to ensure you can actually stop the vehicle and the trailer. This is one of the areas most overlooked. One of the issues is, if what you're towing exceeds the weight of the vehicle towing it, stopping becomes an issue. You'd need to ensure there are excellent trailer brakes and a trailer brake controller installed in your vehicle. This will help control the vehicle and trailer, but depending on the load would still be worrisome.
All-in-all, this might seem doable. With all of the upgrades, you are still looking at dropping a ton of money into a vehicle when in reality you'd be better off just getting a more robust vehicle to do the job. The upgrades I've laid out here are not exhaustive, just what I've come up with off the top of my head. If you are not able to do the work yourself, it's going to cost some money to make it happen. IMHO, if you want to do this, you should trade your pick-up in on a 3/4 ton truck or maybe even a 1 ton. Your pocketbook will thank you with greater safety of yourself, your passengers, and cargo.