Let me give you a moment to finish recoiling in horror at what lurks behind the keys in that photograph. Then another for the realization to sink in that your keyboard probably looks the same.
It is very easy to remove the keys from a computer's keyboard. I'm using a screwdriver in the picture, but that is overkill. All you need is a slight bit of leverage. A dinner knife would cover it. On some laptop key pads, your finger-nail will do it.
Just find a corner, slide your knife or nail underneath and lever it up. The whole key should just pop right out. If not, then try another corner. Be gentle, as the mechanism holding them in place is plastic. Too much pressure and you'll have it snapping.
Once they're out, then shake away the gunk; or use your brush or vacuum cleaner on it. By all means use wipes or cleaners, but be aware that each gap there is exposing the rubber membrane. Try not to get that wet.
You can clean your keys in a sink with some soapy water. They're only plastic. But ensure that they're bone dry (perhaps leave them airing for a day to make absolutely certain) before slotting them back into place.
Oops! You don't know where to put them back! The wise people take a photograph of their keyboard before removing a thing. But there's a standard lay-out. The image below should point to the most common keys.