14/3 to run a small 1650 watt Charbroil electric patio pedestal grill
Most consumer appliances are made to plug into a standard 15A outlet.  Your wire (14/2) and breaker (15A) are both suited for this, and your grill uses (1650w / 120v = ) 13.75 Amps.  That all checks out.

GFCI outlets are required in exterior/wet locations, and a weather proof box is a good idea, so you have the right plan there as well.

I think the only other thing you didn't mention is how you are going to run the wire.  If it's outside, run it in the appropriate conduit and use the correct connectors (may depend on exactly how you need to run the wire), and you will be fine.