I am wondering if I am able to play Doubling Season on my opponent, and if I am not able to, then is there a way to put it into my opponents battlefield.
You can't "play" Doubling Season "on" any player, since Doubling Season doesn't target anything (it isn't an Aura spell, for example, and with certain limited exceptions, no other permanent spell can have targets [C.R. 115.2b]), and where Doubling Season says "under your control", or "permanent you control", the "your" or "you" means only Doubling Season's controller, not any other player (C.R. 109.5).
It is outside the scope of the forum to suggest cards that let you put Doubling Season onto the battlefield under an opponent's control, or otherwise let you give control of Doubling Season to an opponent (by default, a player that puts Doubling Season onto the battlefield does so under their own control [C.R. 110.2a]).
It is outside the scope of the forum to suggest cards that let you put Doubling Season onto the battlefield under an opponent's control, or otherwise let you give control of Doubling Season to an opponent (by default, a player that puts Doubling Season onto the battlefield does so under their own control [C.R. 110.2a]).