If you are confident that the replacement screen works (tested in another phone), the problem is most likely located on the logic board itself. I would start by checking the LCD and Digitizer connectors to see if there is anything wrong (broken or unsoldered pins, debris, missing surrounding components etc.).
However, the iPhone 5c is known to suffer from micro-fractures due to the weaker plastic housing. These are quite difficult to see and usually require magnification in the 20X range and up. This also requires removing the logic board and unsoldering the shields. There may be a problem with the Touch IC's.
Depending on what part of the world you are in, doing logic board repairs on a 5c is usually not worth it financially, unless you have precious data you want to recover. Try my suggestions above and if they don't solve your problem, look for a repair shop that does micro-soldering repairs. They can troubleshoot the problem and tell you how much it would cost. Then you can decide if it is worth it or not.