You can do that by downloading Origin, installing that, logging in with the account you used to install the game with the first time.
Then go to the 'My Library' and install The Sims 4 through there.
For your save games please make a back up: http://sims4.crinrict.com/en/2014/09/backups/
I install TS4 on new PC (DVD version) , logged to Origin ID account and then start TS4 installation. Then shows me message: This code has already been used. It is necessary to uninstall TS4 game on previous poor PC or what?
If you need to put in the code, aka activate the game, you're logged in the the wrong account. If you're still logged in on your old computer, please go there and check in the lower left corner of Origin that you're logged in to the same account (Your ID should be there).
On your new computer, the game should already be in your library. In the Origin Client, click My Game Library and see if The Sims 4 show up.