How do I reinstall the sims 4 on another pc?

Use the account you used to install the game in the first place. The code is attached to that account.

---

I'm trying to register my game but it says the serial code is invalid/already used

You have already registered the game yourself and are trying to reinstall the game
The game does not need to be registered again. Login in the Origin Client with the Email/PW you first registered the game on and download and install the game from there. The game only needs to be registered once and you won't need the code again after your initial registration. This is true for both the DVD and the download version. You can skip the activation step on reinstalling.

Source: http://answers.ea.com/t5/The-Sims-4/FAQ-Serial-Codes-Game-Activation-and-Download-Issues/td-p/4228232