My iPhone 5 is fully charged but won't turn on?

It only turns on when its plugged into the charger. When i take it out it shuts off and doesn't turn on, even when its fully charged.

When i try to turn it on, without the charger, It will show a white screen with the apple for about 5 seconds and then go blank and not turn on. Acting as if it was dead. Should I bring it to the apple store, or buy a new charger/battery? I have no idea why its not working.

(no warranty i bought it online) (its a verizon phone if that helps)

Change the battery