I have an old Hp pavilion 521c that I am trying to get up and running but every time i try to power the computer up, it shuts off in 2 seconds, and after a while it doesn't even turn on. Is there any way to fix this so i can get to the boot screen?

could be a number of things, take a look inside the case and check the fan isnt clogged up with dirt, and look on the m/b for any blown or bulging capacitors
Capacitor plague - Wikipedia, the free encyclopedia should help to show you what to look for, and do you have any spare parts you can use to swap out and help diagnose problem, first thing i would try apart from checking there are no loose connections is swapping power supply for one you know is working

Pretty basic - but it has happened to me, especially on an old machine - have you checked the power button isn't sticking?

It certainly sounds like a power problem, and the motherboard shutting down to protect itself.