Nothing to it really you just need a medium and some fish or fresh water aquatic life that eats and poos. I am using hydroton but rocks and gravel will do. Start with a couple little fish to start the microbes building in the medium plant something and wait for the seed to pop. At that time add more fish slowly a couple at a time a week giving time for the filtration to take affect from the nitrites getting converted to nitrates. You can pee in the water as well to start the microbes.
Here are some videos
Simple explinations
http://www.youtube.com/watch?v=FwMEulvJ2Ps&list=PL8DD9940795FB8097&index=1&feature=plpp_video
http://www.youtube.com/watch?v=DqS9ZXUPz2k&list=PL8DD9940795FB8097&index=8&feature=plpp_video
Bigger with Will Allen he is basically the poster child for Aquaponics
http://www.youtube.com/watch?v=ozvrp_uTH98&list=PL8DD9940795FB8097&index=28&feature=plpp_video
Here is how to make a bell siphon if you decide to go that route. I am just using NFT bell siphons while not hard to set up are problematic at times. I find balancing the water flow is easier and just as affectve.
http://www.youtube.com/watch?v=PLbOW1e9Ym8&list=PL8DD9940795FB8097&index=2&feature=plpp_video
Here is an example to build from plastic storage I find them on Craigslist for about 100-150 used but any food grade container works.
http://www.youtube.com/watch?v=WYFM7J_TpTU&feature=related
Basically once the microbes are built just feed the fish. The ratio is a minimum of 2 gallon of medium to 1 gallon of water so 2/1. I have done it with 5 gold fish 2 koi and a algae eater.
When mine was outside I had a 55 gallon plastic container that you see holding water on the side of the highway exits to stop impacts in an accident and (2) 55 gallon drums cut in half.
Bad side is my dog will not leave the pumps and air stones alone so I moved them inside until I can create a green house.
If you want a closed system so you do not have to buy the food for the fish you can breed and feed black soldier flys from you garbage or even dog poo. The larva eat the dog poo and no poo is used as fertilizer so it is safe source of protein.
The cool thing about soldier flys is that they give off a scent that regular house flys do not like so they actually repel them. Soldier flys only eat as larva and as adults they only want to breed so they are not a nusiance.
http://www.youtube.com/watch?v=qaoVBlS8iVA
The other alternative is to grow duckweed which is bascally a moss they feed on in the wild. As mentined in one of the first videos. I just feed them fish food which is basically ground up fish so it is emulsion fish fert as well.