My iPhone 5 got hot, battery drained fast, and now only flashes apple logo when plugged in?

So the other day i was on my way to school listening to music on my iPhone 5 and suddenly it started getting extremely hot, like so hot you couldn't hold it. Then a warning came on the screen that said i need to power my phone down because it is too hot. So i did and then i waited till it wasn't hot anymore and turned it back on and it was at 65% when before i turned it off it was at 98%. Then the phone got hot again and the battery percent dropped extremely fast so when i got to school i plugged it in but instead of it charging it, it went down. It recognized that the charger was there but i wouldn't charge. So then it finally died and when i plugged it in i just flashes the apple logo on and off. And i thought it was the battery or something so i bought a new battery and the phone turned on but wouldn't charge so eventually the phone died again and repeats the flashing of the apple logo. This is extremely frustrating and i just want to know what's wrong with this stupid phone.

Mine never worked properly again and everytime I called Apple they couldn't help me other than to tell me to pay more for new hardware. I'll never buy an Iphone again.