My iphone 5C restarted then wouldnt come back on, hours later it came up with connect to iTunes, so i did and it wont update or restore my phone. What do I do?

I was using my phone on Facebook whilst on FaceTime to my friend and my phone randomly restarted, so I tried to turn my phone back on and it came up with a fuzzy screen (pink, yellow and blue) on the apple logo across the screen. I turned it off and it wouldn't turn back on. I left my phone over night and it turned on in the morning to show 'Connect to iTunes' which I did and it said I needed to update it. So, I downloaded the update to my laptop via iTunes and it wouldn't update my phone. My last option being Factory Reset my phone, which I then plugged my phone into my laptop to Factory Reset my phone and it did all of the loading but when it finished it says my phone isn't recognised and cannot restore or update my phone. I am not sure what is wrong with my phone or what I should do?

Any Suggestions?