You can, but I wouldn’t recommend it
There’s nothing to physically stop you trying to overclock your PC’s processor while using the stock CPU cooler.
The only thing to bare in mind is that the cooler supplied with your CPU is only designed to keep it running properly at it’s standard speed.
If you overclock it you’ll be generating more heat and you may find that it’s more heat than your stock cooler is able to get rid of.
By all means try, but follow these steps:
1) Research what the maximum operating temperature of your processor is. These figures are available on the paperwork that came with your processor, or from Intel and AMD’s websites.
2) Install HWmonitor which will allow you to keep a close eye on your processors temperature during overclocking. If the temperature of your processor gets near to the maximum operating temperature, you’ve gone too far.
3) Only increase the clock speed of your processor in small intervals (100MHz max). That way you’re not going to overdo it and damage your CPU.
Each processor is different and the amount that you’ll be able to overclock it will differ greatly from unit to unit.
It’s also worth bearing in mind that running your processor hotter will shorten its lifespan, so consider upgrading your cooler for £30 rather than buying a new processor for £100!