You want to start with a 5 or 6 inch net pot. (some places call them orchid pots) It's probably not the best idea to put peat into the solution, that being said if thats what your plants are in, you don't have much choice. Next time I'd use rockwool plugs.

Now put some of your clay pellets in the bottom of the net pot, maybe an inch or so deep, now in goes the peatpot/plant and then fill the net pot the rest of the way with clay pellets.(clay pellets should be surrounding the peat pot)

Now get a 5 gal bucket with a lid, cut a 5 1/2" hole in the lid (so that the net pot goes all the way into but doesn't slip through the lid to your bucket)

Now take your bucket and drill a hole (just big enough for the aquarium pump tubing to go through) about an inch from the top of the bucket.
Slide the tubing through the hole, put your airstone in the bucket and attach it to the tubing, hook your pump to the other end (I personally like to duct tape the pump to the side of the bucket but this is optional)

Now fill the bucket up with nute solution to where the bottom 1 inch of the net pot is in the solution (this is only until you have roots coming out of the net pot) it should take around 4 gal. give or take.
Once you have roots hanging down the solution level should be slightly below(an inch or so) the bottom of the net pot.
And bingo your growing your very own tomatos

There I think I've managed to make a very simple concept about as complicated as I can, anyone else want to try to muddy the waters a little more?

Good luck. I just harvested my first crop using this method, and if you've only grown in soil, you'll be blown away