There are actually some variables which will determine if an upgrade is needed but the number one way to find out is to run a program such as CoreTemp and stress your computer by running another program such as Prime95. While Prime is running, your computer will reach it's highest running temperatures. (CoreTemp and Prime are both free). Report back with the temperatures reported after running Prime for at least 20min.
Keep an eye on temperatures during the test since, if the stock heatsink cannot handle the heat (your main question I believe), the temps could go too high (I shut down Prime and look for lower clocks if I reach 80C during stressing)