Hi! I have been having a similar issue with my iphone 5. (I am using my parents ancient windows xp media center to try to fix)
1. Apple logo flashed when i plugged it into wall charger and wouldnt charge at all or turn on.
2. when plugged into computer would not do anything. Couldn't be recognized(computer didnt even say it was plugged in, even in device manager) and would not even flash the apple logo while plugged into computer.
Tried all of the aforementioned steps to no avail. Was just about hopeless... Then I ran across an article describing DFU mode.
NOTE: If you are worried about any photos contacts etc on your phone DO NOT CONTINUE. TAKE IT TO AN APPLE STORE.
That being said I was able to put my phone in DFU mode by plugging in the phone to my computer. With black screen and no power indication on the phone, I held down the wake button and home button for 10 secs. Released the wake button and continued holding down the home button for another 10 secs and then released it as well. Suddenly my computer recognized the phone in dfu and was recognized in itunes and I am able to now restore it(Aka erase everything and reinstall the firmware). I have not finished it yet but will reply back once it is done and will let you know if it works. The screen has stayed black the entire time.