Do you use your laptop on your desk? A laptop isn't actually meant to be used on your lap or on blankets.
Applying thermal paste on a laptop CPU is going to be a difficult feat, you'll need to disassemble it open to the motherboard which in itself is pretty delicate and can leave you with a paperweight. And it also technically voids your warranty (but you don't have one here).
Since you don't have a warranty I'd try to figure out where the problem comes from - it probably isn't thermal paste to be honest, it's more likely to be a fan failure, you'll be amazed how quickly your laptop heats up when the fans stop working, even under light use. Furthermore, if the thermal paste was indeed the culprit your CPU would probably have taken some heavy punishment by now and the laptop would likely be dead. Thermal paste as a fluid compensates for surface irregularity, and if it were to fail, the little CPU parts that "stick out" would quickly melt.