Nexus 5 turns on and inbetween boot up and home screen, shuts off at random times

The phone appears to turn on normally and at cold boot it manages to go to the home screen. I can be there for a few seconds, then it'll randomly decide to shut off. I'm not sure but the temperature seems high closest to the CPU and screen feels hot to the touch when trying to keep it on. Should I plug in a brand new oem battery? I have a knock off battery and it worked fine on my other phone except the capacity was much lower. I thought it might be temperature related so I tried making an aluminium heat sink and thermal paste for the cpu, but nothing changed. I will link to video in a bit. Also the only way the phone responds after is by plugging in the phone cable to any power source as if the battery was taken out. Don't judge the vertical framing it just happened. I tried to replicate it again all normal like and the phone stops at the 4 color dots and the phone shuts down as if the battery was removed. It doesn't respond until plugged into usb power. HALP