Once you are able to do "online" missions, select the Online Contracts App from the game's phone menu, and choose the Tailing option. (Note: Sometimes certain actions you take will prompt the game to offer you an online tailing mission, so you can accept that when it happens as well).
It can take a minute or two or more before you're connected to another player. Once you are, you'll be teleported to their location, and the tailing will begin.
The other player may or may not be aware that somebody is tailing them. If you open doors or raise lifts via hacking and they notice it, they might realize somebody is tailing them. Or if you're watching via camera, and the other player tries to hack the same camera, they'll see "In Use" which is a tip-off too. Also, I think that the game begins to play music when a tailing begins (an in-game instrumental song, when no other music is being played), and if that's true, it may also tip off the player to begin looking for you.
At least twice, my game started lagging down pretty bad when nothing else was going on, shortly before I realized I was being tailed. So if you have a poor internet connection, it might also be a tip-off that they're being tailed.
Otherwise, they probably won't even be aware what's happening. You'll see the player doing whatever it is they're doing in the game (standing still to hack cameras, chasing somebody, etc). You just have to stay out of sight and not get spotted (profiled) until your "watch" meter is at 100% to unlock this trophy. You must keep a line of sight to the player for the meter to advance. Watching via cameras counts too, but can potentially tip them off if they notice it's in use.
When the tailing begins, you may have to catch up to the player or race after them if they're on the move. Just be careful to stay out of sight so they can't profile you while scanning what they believe to be NPCs, until you're in a position to keep them in sight without being seen yourself.
If you do get spotted, simply try again.
When I think I'm being tailed (via one of the tip-offs I mentioned), I will usually try to find a place to hide (up some stairs and behind a secure spot, or inside a garage after closing the door, etc). Take cover and wait and watch, and often the other player will come to you, making it easy to profile and kill them. So if you're trying to tail somebody and they disappear off to a hidden area, be careful going after them. They might be waiting for you.