Cleaning Type K thermocouples - What is safe?
It is clear from the picture that the actual thermo junction is contained in a metal sheath. If you saw the actual junction, it would look like two thin wires welded together where they meet. As such, you can aggressively clean (but perhaps not submerge) the probe tip with any means that does not harm the metal jacket. I would probably choose steel wool to give it a quick buffing up.

On the other hand, in a grill application, it would take a tremendous buildup of byproducts before having any real effect on the application. The normal buildup will slightly slow the thermal response (seconds) but it will not cause an overall temperature error.