My iPhone 5 will died, and now will not turn on even though its charging?

My iPhone 5 died yesterday, so I immediately plugged it in. It was plugged in for about an hour and would still not turn on. I tried resetting it but nothing happened. I then cleaned out the charging port and got a ton of lint out, and then when I plugged my phone in, it turned on and started charging within a minute.

Well, today the same thing happened. Its been 2 hours and my phone will not turn on. When I press the on button, or the home and on button to try to reset, the symbol of the battery life shows up, but I never get the phone to turn on. I DID get it to turn on once, when I plugged it into my computer and loaded iTunes, but after 3 seconds of it being on, it died again. I tried closing and opening iTunes back up, plugging and unplugging, nothing. I've tried everything the internet has said to do, but even tho my phone is capable of turning on (has been charging for 2+ hours and turned on for 3 seconds) it will not. Any advice?

Is your charger broken? That has happened to me a few times.

Try to connect with pc or laptop it givs respons or not… If it response then it will charge too otherwise use export help

Also: (John)- My charger is nearly brand new. I've also tried another charger, get no results from either

SECOND UPDATE: It just turned on after being plugged in for 2 hours, but its only at 5% and is still dying even tho its been charging for over 2 hours?

Have a new battery installed.

That's happening to me but the screen is black no matter what