If your motherboard is newer and of decent quality, the BIOS should have a section that will tell you the current CPU temperature. You should check this by soft-booting after the computer has been running for a while, to get the most accurate reading.
I don't know offhand what the maximum operating temperature is for any particular CPU, but if you have some basics in-place (and you don't overclock the processor), you should be fine.
Here's what I recommend for CPU cooling in pretty much any standard PC:
-Your cooling unit should be at least standard size (i.e. made or sold by the CPU's manufacturer for that CPU). You can also get one with bigger/more fins and a bigger and/or more powerful fan. Although I strongly recommend against overclocking your processor, if you actually DO it, you must have a better cooling unit than standard. Just make sure it will fit tightly on the CPU when installed.
-There must be heat sink grease (a special, usually white paste) between the CPU's contact point (a smooth flat surface on the CPU) and the cooling unit. There should be no bubbles in the grease. This is essential to ensure your CPU doesn't overheat. The grease both helps transfer heat from the CPU to the cooling unit, and helps hold the cooling unit in place on the CPU. If you unhook the clamps holding the cooling unit to the CPU and it is still stuck there, then you probably have heat sink grease in there.
-Any extra chassis fans you can add also help, but aren't completely necessary. Some cases and chassis cooling kits also include hoods that direct incoming air directly from one vent, through the CPU fan, and back out of the case. That's also helpful, but not 100% necessary.
Hope this answers your question.