Good Tennis Courts available in Shinagawa, and Ariake.
Takanawa Tennis Center in Shinagawa (and they have indoor tennis courts)
Ariake Tennis no Mori in Ariake
As a matter of fact, I'm a novice tennis player, and it has already been several years since I played tennis last time.
When asked for advice about the topic, as far as I am concerned, Takanawa Tennis Center in the Shinagawa Prince Hotel will be a good choice for its convenience though it might be a bit expensive.
I have looked for more information about Tennis Courts in Tokyo which will be run by public services, then I came across "Ariake Tennis no Mori Park", which I think looks pretty nice with 16 courts available, and the cost is just reasonable for its convenient location near Odaiba, the Tokyo Bay water front area. This one will be also a great alternative.