How frequently are airplane windows cleaned?
I clean them whenever they are dirty to the point of affecting ability to see traffic or cause distractions. 

Bugs, dust, and dirt can make it very difficult to spot traffic, birds, and obstructions, especially in hazy weather. 

It is the pilot’s responsibility to “see and avoid” and clean windows play a large role in this critical safety area.

Depending on how bad contaminants are (i.e., if it’s bug season) the interval for cleaning could be as often as every flight or as little as once every few days while continuous flight operations are in effect.

Windows are cleaned with either water or special chemicals for cleaning plexiglass with microfiber or non-abrasive cloths. Also of note, human fingerprints are ideal for cleaning aircraft windows with water as they are soft and do not scratch, but the ridges pick up the dirt and easily rub off even sticky contaminants [bugs]. Aircraft windows are relatively soft and scratch very easily. All cleaning motions are done in the direction of the airflow to prevent hazing of the window from scratches... no circular scrubbing - only in the direction of air flow. Passenger windows rarely get dirty. It’s the ones for the cockpit that are directly exposed to the frontal airflow that get the worst of it.

I usually use water, my fingerprints for scrubbing, and clean microfiber cloths for drying to prevent water spots - which can obscure vision as bad as dirt.

Here is a video of the process.