Many stoves today have a ventilation hood above the burners. These hoods ventilate heat and smoke from normal cooking and usual have some grilles in the design of the hood. This part of your stove gets very greasy from cooking. Not only is this an appliance in the home that needs a regular wipe down after cooking, but many of these grilles can be removed from the hood and are available for a much deeper clean. This will be necessary because the amount of grease that accumulates creates a dirty environment and a potential fire hazard.
The first thing to do is check the owner’s manual for the proper instructions for your specific brand and model. The following are some general guidelines. Most hoods have removable parts like baffles or even filters. These can be removed and washed in warm soapy water. Some models have charcoal filters which would need to be replaced versus just being cleaned.
Now you want to sponge down the inside and outside of the hood. You want a solution that can cut through the grease to essentially wipe everything down. If your hood is made with stainless steel, you should use a specific stainless steel cleaning product. Spray the hood with the cleaner, and with a microfiber or lint free cloth, wipe down the surface moving back and forth with the grain.
Here is a quick video from Danny Lipford on some of the basics.