There are a great many different types of aquarium filters. You can likely keep using your existing aquarium filter when setting up your fish tank as an aquaponics system. In fact, you may need to keep using it depending on the system you select. Your hydroponics system is a living system, and as such it's performance as a filter is ever changing, and your 'aquarium' filter will be needed to pick up the slack!
Aquarium filters operate on 3 different levels. Mechanical, biological, and chemical.
Most natural bodies of water are turgid to some level - that is, they have a very fine sediment level that reduces visibility in the water. This is considered a bad thing in aquariums, as we want to be able to see the back of the fish tank and all of our little fishy friends. Mechanical filters remove the solids from the water. This can be fish poo, excess fish food, broken off bits of aquarium decorations, dead fish, random stuff that falls into the tank - anything you can see. This is a huge factor in keeping water clean looking.
Chemical Filtering is the removing of 'chemicals' from the water. This could be anything from the trace amounts of lead found in residential water systems installed before the '70s due to the lead in the solder used to join the pipes, to the tannins released by that beautiful piece of driftwood in your aquarium. Probably the biggest chemical an aquirist is interested in removing is Chlorine. This is added to municipal water supplies to reduce the biological activity in the water - keeping it safe for humans to drink. Fish really don't take well to chlorine though. It can be bad for plants grown in hydroponics systems too. I also suspect that chlorine is bad for the biological filtering system in an aquarium, but that's just my theory, and I haven't found or done any research to prove this.
Biological Filtering is using 'good' bacteria to break down the ammonia released by the fish's gills as part of their respiration process into Nitrites, then Nitrates. (Check the order of reduction). Establishing this bacterial colonization is what all the coarse foam sponges, fancy highly porous sintered glass and 'cycling' a tank is all about. Ammonia is bad bad bad for fish. Nitrates & Nitrites are fertilizer for plants. Regular plant fertilizer is sold using a NPK score, where the N is nitrogen!
You may need to adjust what media you put into your filter so it works better in conjunction with your hydroponcs system. The aquarium filter system will need to do all the mechanical filtering. It will be responsible to clean out all of the solids from the water. If not, these can end up getting lodged into your hydroponics system - what a stinking mess! You will leave the biological and chemical filtering to the hydroponics part of this new aquaponics system, while keeping enough 'reserve' capability in the aquarium filtering.
There are many types of filters used in aquariums, and most can be made to work with some form of hydroponics or another. From simple air-stone powered sponge filters and under gravel filters through filters that hang off the back of a fish tank, to external canister filters and aquarium sumps. They call can be made to work with some type of aquaponics. You just may need to get a little creative to make your own aquarium aquaponics filter from what you already have.
Air pump powered sponge and undergravel filters work by bubbling air in a tube. As the air rises in the tube, it brings water up with it, creating a water flow. This water flow either pulls water through a sponge or through the gravel at the bottom of an aquarium. These style filters can work with water hydroponics or bubble style systems where the roots hang into the tank directly.
If you add a powerhead - an electric pump - to the top of one of the air bubble stand pipes, you can now use one of these types of filters with many other types of hydroponics systems.
Over the back filters can be a bit more challenging to use directly into a hydroponics system, but it can be done. If you place a shallow grow bed directly over the top of the aquarium, you can actually just hang the filter on the side of the grow bed in a way that it pumps water from the tank and spills it into the growbed. They make models and accessories with extra-long pickup tubes for tall aquariums if you want to have your grow bed a little ways above the top of the aquarium. With these though, the higher the top of the filter is above the top of the tank, the less water flow the filter can create, and if it's too tall, the filter may not be able to prime itself when the power goes out or even pump any water at all.
If you have one of these running on your aquarium already, it's fine to leave it running and use an external water pump. This style filter would be a good thing to have if you run an intermittent type ebb and flow bed or turn off the water pumps at night as some people do. They would help keep the water quality good for the fish.
External canister filters are probably your best bet for small to medium aquarium filters to use in aquaponics. They don't cost to terribly much more then just a water pump will, and have the added benefit of cleaning the sediment from the water before you pump it into your grow bed. I honestly don't know why more people haven't picked up on using these style pumps - maybe they just aren't familiar with them, or maybe the extra price scares them away. These are very simple to use, hang the inlet site on the side of the aquarium, and the outlet side on the growbed. Because the filter is usually placed under the aquarium, when the power goes out, they won't loose their prime. Because the outlet is in the growbed and not in the aquarium, if the power does go out, some of the water in the hose will flow backwards through the pump until the water level on the growbed side equals the aquarium water level. There is a small likelihood that some filtered solids may flow back into the aquarium. But if you kept the filter pickup hose longer then the hose length difference between the aquarium and the growbed, you shouldn't have any problems.
Sump filters are the best type of filter to have for aquaponic aquariums. They are usually also the most expensive and take up the most space. These are designed in such a way that the main aquarium overfills and the overflow is piped to the sump. Traditionally, the water is then pumped from the sump back into the aquarium. For an aquaponics system, you simply pipe the return to the growbed(s) and let them flow back into the aquarium. The nice thing about the sump system is that the water level on the aquarium always stays the same level.
There are a few things to make sure of when using a sump style filter. The aquarium overflow needs to be able to handled the peak water flow coming from the grow bed. If your using a bell siphon ebb and flow system, you need to be able to handle the 'flush' from the grow bed. The other thing to watch for is that you don't over-fill the system when it's running. It would be very easy to top off the water level on your sump, and end up with more water in the system as a whole.
When you first set up a sump filter, fill the aquarium and sump so they are both full. Then run the system for a little bit, and then turn it off, letting all the water settle back into the sump - top it off at this point. Some water will pool at the very bottom of the growbed, in piping, etc. that won't drain out into the sump. Now turn everything back on and watch it closely for a couple of cycles. The water level will drop in the sump. Mark the lowest point you see it drop to. Only top of the water levels when it's below this mark, and only fill it to that mark - no matter where you think the system is in it's cycle. Now you shouldn't have to worry about the sump overflowing if the power ever goes out. If you make significant changes to the system like adding or removing a grow bed, you will need to re-calibrate your fill line by starting this process over again.
There are many specific aspects of filtering to think about.
The material that most commercial filtering systems use to keep aquarium water crystal clear is Activated carbon. This works great in aquariums. It removes water impurities, heavy metal ions, all kinds of stuff. Unfortunately, aquarium filters using activated carbon can't be used in aquaponics systems as they remove the very trace elements and minerals that the plants need!
If you have really icky water for some reason, like you just did a large water change, feel free to use activated carbon to get the water crystal clear again, but stop using it as soon as you can. You may want to dose your tank with hydroponics fertilizer after you removed the carbon filter to replenish those trace minerals.
I've recently read about the natives of the amazon jungle creating their own 'super dirt' using charcoal called Terra Preta. http://en.wikipedia.org/wiki/Terra_preta I suspect that if you did want to use activated carbon in your aquaponics system, you should move it to the grow beds, where the plants can root into the carbon and access the organics that the carbon collects directly. I am trying to find some real research on this application of activated carbon in an aquaponics system, so if anybody has a link, please let me know! I will likely do some 'real' research the old fashioned hands-on way and build two otherwise identical systems one with the carbon, and one without, and see if it does indeed work well.
Chlorine is some pretty nasty stuff when it comes to aquariums. It kills fish. Even worse yet for aquaponics systems, is that it kills the good bacteria we work so hard to develop.
The 'usual' treatment for chlorine is to use those little bottles of chlorine drops. If you read the labels on them, many also say that they remove 'heavy metals'. While this is normally a good thing, the drops will precipitate out most any and all metal ions from the water. This is bad for hydroponics systems as we want many of those trace metals as magnesium, manganese, molybdenum, copper, etc. are all micro-nutrients needed for plant growth.
That, and I don't know what's in those drops. Many also say 'for ornamental fish only'. Meaning, not for fish for human consumption (not that we will be eating the fish out of our aquarium). Meaning, we don't want to grow food plants in water that's been treated with this stuff either.
How I treat chlorine for my aquariums is I age the water. I simply let it sit for a few days before I add it to the tanks. This allows the chlorine to outgas into the atmosphere. If I am in a hurry or have a deep bucket to let age, I will aerate the water with an aquarium air stone. This is a simple, safe, and organic way to treat for chlorine.
I will let you in on a little secret. Vascular plants - the types of plants we are growing in our hydroponics systems - are more efficient at using these fertilizers then algae is. So by keeping a balanced rotation of plants and a balanced mix of nutrients in the hydroponics system algae shouldn't be a problem.
Now, algae can grow much faster then plants, so if you dose your system with massive amounts of phosphorus or potassium or something, the algae will grow to consume that extra dose faster then your regular plants for a short while. If you do any augmenting fertilization, smaller doses more often is better then larger does less often as far as algae growth is concerned.
If you are getting algae growth on the surface of your growbed media (if you use a system that has it) your water is coming up too high into your growbed. Lower the water level a little bit so the top surface of the growbed doesn't get damp.