Algae is a very big problem in hydroponics. It competes with the plants for nutrients (it eats the same thing plants do). It can grow on the roots of plants which inhibits their ability to absorb water and food.

Plants in soil can benefit from being watered with water containing algae, but this is because soil allows the algae to dry out and the microbes native to soil can eat the algae, breaking it down into food for the plants. In a hydroponics system such microbes don't exist unless specifically introduced. (A nice idea, but just one more thing to spend money on. It's easier and cheaper to simply prevent algae growth.)

Fish and hydroponics together is called aquaponics. However, this is normally done by pumping water from a tank/pond into a hydroponic system and allowing it to return to the tank/pond. The plants effectively clean the water, the fish feed the plants.

However, this type of system can be exceptionally complicated. Too many fish and your plants will be overfed and unable to clean the water sufficiently. Too few fish and the plants will be underfed. It has to be properly balanced, which can be tough to do.

There are massive farms that do this kind of thing. I've read about one in particular that raises some kind of bass and uses the water to grow lettuce and some herbs. They sell both the fish and the greens to markets. Virtually zero run-off and all they have to put in is food for the fish.

This might be some help.
http://www.youtube.com/watch?v=FwMEulvJ2Ps

That's a bit more complicated than many of the aquaponics systems I've seen, but very elegant. I particularly like that she's using the system to grow the food for the fish, too.

Tilapia are ideal fish for aquaponics because they are bottom feeders - they help keep their tank clean by eating algae.