are you saying if the CPU comes with some kind of stuff (for lack of a better word) to use it and if not to use this paste I bought?
In the old days, heatsinks were shiny metal where they contacted the CPU (also shiny metal). That direct metal to metal contact was actually metal to air to metal since even apparently shiny metal has lots of microscopic pits. The thermal grease/paste/compound is needed to bridge/fill those microscopic pits to improve heat conduction. Nowadays, most CPU heatsinks are supplied with a patch of stuff stuck to them that does the same thing. It usually does not look like grease, but rather like a patch of something. If you look at the CPU cooler where it contacts the CPU and you just see bare metal, then you need the stuff you bought. But if there is a patch of stuff there, you do not. I expect there are some cooling afficiandos out there who will remove the supplied patch of thermal coupling material, sand down to bare metal and then apply some mysterious silver containing compound that cost an arm and a leg to get to an imagined optimum transfer. If you do not remove the supplied patch, then you will probably make the situation worse by using grease on top of it, not better.
btw some of the fancy silver containing heatsink compounds turn out to be electrically conductive as well, and if not really carefully applied can cause short circuits.
Your CPU spec sheet will give you an operating temperature range, and your motherboard bios, or some utility software will give you an accurate enough reading of the CPU temp to see if you are cool enough. Unless you are massively overclocking, or your CPU fan fails, it is unlikely that you will fry your CPU.