Sounds like a faulty digitizer. Is it responding to your touch at all? If not, you can check the connections, maybe clean them with some isopropyl alcohol on a cotton swab.
If it still doesn't work after that, you'll probably have to replace the digitizer. You can choose between this: iPhone 5c LCD and Digitizer which would require you to switch out the home button and front camera, or you can buy this: iPhone 5c Screen with the home button and front camera included.
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