So, I would avoid the string/hanging concept if possible. Far too much potential for movement. You want a solid, firm foundation for the part to be painted.
The goal will be to elevate the piece high enough to get a clean shot with the spray can, yet, still allow you to close the box to minimize the effects of random dust. Since I have the lid off anyway, the pics enclosed below are a simulation of what I mean. In this way, you have complete access to the piece to be painted while not allowing any one edge to be obscured or obstructed from a clean shot.
I have to assume, since you went to Home Depot for supplies that you are using Rustoleum spray. My weapon of choice. With that paint, you can apply a light coat and re-coat in about 30 to 45 minutes with another coat. Don't go too much longer, otherwise, wait a day (or at least overnight) and lightly (and I mean lightly) sand and spay another coat. This is always your fall back position. Let it dry, lightly sand with no less than 220 (320 might be better) and re-coat until the texture and depth of color is what you are looking for. This is true regardless of color. This is the beauty of DIY, just take your time and do it as many times as it takes until you are happy with the results. You cannot rush paint. It takes as long as it takes. If you mess up or or are unhappy with the results, sand/strip it off and start again.
Good luck to you! Looking forward to the end the results.