Usage of many of the public courts is allocated based on lotteries. You'll play weekend or evening tennis more regularly by joining a group whose members all enter the lottery together. If you enter just yourself, you may get to play once or twice a year; if you enter as part of a group of 20, 40 or 60 persons, you may get to play once or twice a month. This approach does result in doubles play, though.
(Getting weekday daytime playing time is much much easier, but you'll still play more if you have a group of 4 to 8 like-minded friends with whom you mutually share court times.)
How to join a cooperative? Well, you can start your own. Push your friends to get registered, then you turn in their applications for them. Or attend the lottery sign-up sessions and bravely approach groups you can identify. Or, at the courts, approach other players and ask if they have a group. Don't be shy - at minimum the response will be polite, and you'll likely be pleasantly surprised by how open the players are.
When I attend the Minato ward monthly lottery time-slot sign-up, I see several cooperatives of various sizes in action. I am in the Tokyo American Club's cooperative. There is a largish group of Japanese players whose cooperative "managers" sit right at the desks near the front, have detailed forms, and are very very systematic. Other smaller groups mill around in the sign-up room.