In this case you would use “on”. You wouldn’t use “with” because a machine (in general) doesn’t play back with you, like a person or pet would. However if you were in a multi-player game, you could say “I am playing with my friends on the PlayStation.”

You can say "my children play on a playstation" but, people where I am from (florida) we mostly say "My children play on the playstation" or "My children play often on the playstation.