What you will need?
The one thing that ends up consuming a good bit of electricity is the water pump in the aquaponics system. So it makes sense if you're able to power this pump using solar power. The easiest way is to purchase a low amperage pump. This means that the pump can be powered with very low amperage of electricity and will require small solar panels. A 24 watt pump is ideal because it has enough power to pump water up to an elevation of around 8 feet. For smaller systems that do not require that you pump water to an elevation of 8 feet you can settle for something like a 12 watt pump. The solar panel that powers it will also be a lot cheaper.
Installing the solar panel
The next step is to install a solar panel near the aquaponics system. Ideally, you will want to mount the solar panel as high as you can, probably on a pole or on your home's roof but facing south if you are located around the northern hemisphere. Depending on the type of solar panel you purchase and the wattage of the pump you will need to purchase either one or two panels. The solar panels (in the case of two) will have to be wired in series in order to output the required potential difference.
You may also want to purchase a regular pump controller which will control the electricity from the solar panel over to the pump. A battery attached to the controller will store excess electricity which is then used in low light conditions. Then you will need to hook up the pump to the controller and the controller to the solar panel in order to complete the circuit.
Back up power system
If you live in a place where the seasons change drastically and sunlight is somewhat rare you should also arrange for a backup power supply system. This will usually consist of a long cycle lead acid battery coupled with an inverter or a direct connection to your home's mains. You can either hook up the backup system manually or you can design a system that automatically detects a reduction in volage and switchs to backup in order to power the pump.
Powering your aquaponics equipment with solar power makes a good deal of sense for anyone who is trying to maintain a large system. However, for smaller systems like the one residing in your living room purchasing a solar panel to power it does not justify the cost.