My iPhone 4s won't connect to iTunes or Charge. Read Details?

Last night, I put my phone on the charger and when I woke up, it was completely dead, no charge what so ever. I thought maybe it was the usb cord so I used a different one and the same result, it wouldn't charge at all. I took it to school and tried to charge it using my school computers and it still would not work. When I got home, I plugged it into the computer charger (remind you, it's completely dead) and viola it charged on the battery screen where the battery is red. Then, my phone began to power on but as soon as the phone came on, my computer kind of rejected the USB saying it couldn't recognize the device and iTunes couldn't recognize it at all. My phone was only charged to 4 percent. After a while, iTunes recognized it just in time for me to do a quick backup and then bam, it's not recognizable again. The battery died out eventually and then again, it began to charge and power up, then it stopped charging.

I would try to put it on dfu mode but it won't work because I don't have a good enough charge.
What should I do? Help please.

Added (1). I tried 2 different charges, both using a computer and using a wall dock. I also tried to clean out the usb port on my phone, that didn't seem to help much.

Why don't you use the original charger that came with the phone & plug it in a wall outlet? I don't understand why you have to use the computers to charge your phone?

Let your phone charge through an actual wall outlet, then perform a cold re-boot, you can do that by holding down the top on/off button until the screen shows" slide to turn-off" pops up.
Slide the bar & turn of the phone. After a few minutes.hold down the top on/off button again & let it come back on.

This should automatically reset it & now when you connect to i-tunes, your computer & phone should recognize each other.
Hope that works.