I have a 300 gallon aquarium in my greenhouse with trout and perch in it, and hydroponics shelves above it. The shelves are watered by pumping water from the aquarium to the top shelf, and it drains through the shelves back to the aquarium, and the plants in the shelves use the fish waste as fertilizer and this cleans the water. What plants will be good in the shelves to best filter the water?

Your plants are not going to get enough nutrients from the poop generated by the number of fish you can support in a 300 gal. tank. If you add nutrients to the water, the fish wont like it/

water hyicynth and pond lettuce are good . talk to your local pond installer

I wish I knew. That sounds like an amazing setup. If you have any local specialty fish places (not chain ones), it would be a good idea to ask there. They could give you the recommendations and supplies necessary.