So I plan on using two ducts, one to take air from my room and move it to the cooling coils of the air conditioner, and another duct to redirect the cold air from the window air conditioner back into my room. The window air conditioner will be located outdoors next to my room. I might use fans to help with air flow if the air conditioner's fan is not strong enough. Can someone tell me why this is a bad idea and how to go about constructing it well?
There's really no problem with doing so outside the logistics of it, assuming you use large enough and/or smooth enough ducting. I wouldn't try to do it with flexible dryer hose, for example. 6" smooth (rigid) or 8" flex might suit.
I'd keep a close eye on things to be sure that it's not either overheating or icing up.
I saw this set up you are thinking about doing in 1968 at a donut shop in Pittsburgh, Pa. and it worked okay. (wow how time flies) They did a lot of stupid things then to save a few dollars. The duct work was oversized to reduce friction loss for both the supply air and the return air. Servicing this unit was extremely hard and very expensive since the duct had to be disassembled every service call. After a few years and many very expensive service calls they made the correct decision and installed a different system. Sometimes cheap ideas are just cheap and dumb ideas. "My 2 cents".
You would have to insulate the both the duct from the conditioned space and the duct to the conditioned space since the air in both will be cooler than the outside temperature. You would have to use flex duct and then protect it from the sun, weather and animals.
This installation would be unsightly and interfere with movement outside. The part of the a/c unit which would normally be inside would have to be protected from weather because it is not designed to be rained on or otherwise exposed. You would have to go outside to adjust the controls, unless the unit has a wireless remote. This installation would void the warranty.
Abandon this idea!