This trophy is earned outside of the Animus, when you're walking around with your communicator in the hallways of Abstergo Entertainment. To exit the Animus into this mode, go to the pause menu with  and scroll down to "exit", and then select "exit Animus".

If you press  while walking around, you'll open up a minimap on your communicator. The minimap will show you which computers you can hack nearby. Some computers will be locked behind security clearance doors—Level 1, Level 2, and Level 3—which you can only open after obtaining those levels of clearance through the story-based sequences that happen outside of the Animus.

There are 33 computers available to hack, but you only need to hack 15 for this trophy and the Platinum. You'll be able to hack a total of 15 computers at the end of the third mission, right before you earn  Getting Weird Around Here, after you get Level 2 security clearance.

Actually hacking the computers requires walking up to them and pressing  and then completing a short minigame. You'll get tutorials on these minigames in each of the story sequences, and they're all pretty easy, so you should have no problems with them.