HP boot and heating problem

Hello, I was wondering if someone could advice me on a problem.
I have a hp dv2125nr laptop and have 2 issues with it. 1 is that when I boot it up, often nothing appears on the screen, the computer at least starts to boot up but the screen is black. It doesn´t seem to be connected to the screen lid at least moving that around doesn´t seem to change anything. Then if I turn it on and off a few times, in the end it will start normally. Do you think it is the graphic card or something else ?
Also, the computer heats up alot, and eventually shuts down or freezes (not sure if it is the heat or not). Buying a new fan could fix this right? and if so any recommendations?
Thanks Stefan