If you're not growing too many (1-3), Deep Water Culture (DWC) in a 5 gallon bucket is the way to go.
A water pump with a sprayer is nice to mist the bottom of your net pots until the roots grow into the nutrient solution.
You generally only put 2-3 gallons in the bucket and make sure to keep it well aerated.
Good luck and post pictures

I think I am just going to grow one of two peppers in a container. I will probably use 5 gallon buckets-one in each. I was thinking to submerge the "pot" with the medium a few inches into the water/nutrient solution.
1. I am thinking of getting Dyna-Grow nutrient, but don't know where to get that or a growth medium such as clay or rockwool.
2. I've heard that perlite works too, does it work as well?
3. I've heard that the nutrient solution needs to be change once every week or so because it gets harmful compounds or chemicals from the plant.

Thanks, I'll be sure to keep you updated once my seeds arrive and I germinate them!
Brian

Your pot doesn't need more than about ½" of nutrient to stand in.

Perlite works well. I mix it with wood chips to lower the cost.

Your nutrient lasts longer than a week, but it does get dirty, and if you leave it too long, it becomes smelly.

If you have a pond and fishtanks there's probably not much else you need - and you have several options. I've had very good luck with peppers grown in perlite, with a recycling nutrient drip. I use 2-liter soda bottles, with the bottoms cut out and the tops glued into 1-inch PVC tees. These are connected with cheap 1-inch PVC pipe, which drains into the reservoir. A small pond pump delivers the nutrient solution to adjustable drip emmitters via 1/2 plastic hose. Drip emmitters are zip tied to the top (actually the bottom) of the bottles. Bottles are filled with perlite.