You may remove the dust from the heatsink using a can of compressed air; you can get such product from any computer/electronics store. I honestly do not think the hair dryer will have the power to produce the same air pressure. Cleaning off the dust will help in case it is blocking normal air flow for the heatsink and rotational speed of the fan (i.e., lower than normal fan RPMs).
If the issue persists, (or for a full approach) you may want to remove the heatsink, clean it out of the case, inspect its pins just to make sure they are not broken or bent and integrate it again applying new thermal compound if needed. When properly installed, the heatsink should not wiggle or shake if you attempt to move it; the pins should not come up if you try to pull them up. Please refer to the following guides:
It may not be your case but you may also reset the BIOS in your motherboard as an incorrect setting or overclocking the CPU or memory may cause the processor to overheat. Keep in mind other settings may be reset as well affecting the operating system. For this step, please refer to your motherboard’s user guide.