It's not a good idea to use your grill for a heat source. Too much risk of spillover having a disasterous effect. I purchased a frying grill base at my local grocery store last year, but I've seen them at Wal-Mart. You get a set of hardware that include everything you'll need, with instructions for use. You provide the gas bottle.
The kit has the perforated liner. The liner is a better choice than the "hooks" you refer to.
The kit also has a long thermometer. You'll need it, so keep it handy.
Oil? Peanut oil has the best qualities, in that it has a relatively high smoke point. I've used canola, too. Don't use shortening because it burns way too easily and is hard to control the temperature. The key to selecting the right oil is smoke point. Get the highest you can.
It takes about 4 minutes per pound of turkey, so a 13-pounder cooked in about 42 minutes at 350 degrees F.
As to the size you want, maximum weight will come with your instructions that come in the deep-frying kit.
One piece of advice you didn't ask for: Make sure the bird is completely thawed, and that you've thoroughly blotted out all moisture inside and outside the bird. Any traces of water will cause you grief, because it's the nature of water to evaporate when hitting 350 degree oil. Forcefully, if necessary.
I'd recommend that you do what I did when I first got my turkey frier: Buy a whole uncut chicken and experiment on members of your immediate family before you cook for company.
I guess I wasn't clear, I have the type of burner that is commonly used for this. I have used it previously for large pots of clams. It's something like 70,000 BTI's.
Great advice - worth repeating. No one wants their turkey to go "KABLOOM!" on Thanksgiving Day. :-)