iPhone Apple logo to black screen

Last night my phone randomly became unresponsive to clicking the home or power buttons and was stuck at a black screen. I had another phone call me and my phone actually rang and vibrated, but the screen was still black. Then I looked up what the problem was and saw that I needed to reset my phone by holding the power and home buttons for a while. After doing this I saw the apple logo appear and thought it was fixed, but it is not.

Now the phone will just stay at the apple logo screen for a while, then the apple logo will disappear (but the screen is still slightly backlit), then it will go back to the apple logo a short time later. I let my phone do this until it finally ran out of power but when I plugged it in and charged it, it just kept on doing the same thing. I also tried putting my phone into DFU mode: by holding the power and home buttons for a while until the screen went black then releasing the power button but continuing to hold the home button. However after doing this and plugging my phone into my computer it did not recognize it and my phone just went back into the same old cycle. I just want a working phone back so someone please help me! I don't care if I have to restore it.

[Re-Titled by Host]