The general rule for Assassins during team fights is to pay attention to the enemy's forms of CC, and leap in the moment it's burned. After that, it depends on the Assassin.
Assassin Xin Zhao, or full AD Xin Zhao, generally wants to fire his Ultimate off right away and scatter as many targets as possible. This in and of itself is usually more than enough for Xin Zhao to kill his target as he uses his Q+W to wail away, assuming the target in question isn't hyper mobile. What's important is that his teammates need to be able to capitalize on the enemy team now being separated, and focus on picking them off one by one. As a tip for the Xin Zhao player in particular, he needs to know how to flank correctly, and much more importantly, he needs to tell his teammates to get ready for the moment he ults.
Similarly, Pantheon has the ability to scare people into separating, and once again this is a case Pantheon's teammates wanting to take advantage of the enemy being separated. Pantheon doesn't nearly have the emphasis on flanking the enemy team because his ult can literally drop him on top of the enemy carry when it is necessary, but at the same time, the Pantheon player in question MUST get his teammates to time their engages properly. A team that isn't ready to back up Pantheon when Pantheon falls out of the sky leads to a dead Pantheon.
Nocturne, at least full AD Nocturne, is more about maintaining vision supremacy and catching people who get separated from protection. For example, if you as Nocturne notice the enemy team's Assassin trying to flank your own team, you can cut them off with your ult, block on of their damaging abilities with your Spell Shield, and more than likely kill them off in the ensuing 1v1. Similarly, Nocturne's Ultimate dash is usually fast enough that the moment the enemy uses their CC, you can jump on people, even from extremely far away, and can get on top of them before their CC comes off Cooldown. Nocturne isn't necessarily about fancy mechanics aside from timing his Spell Shield, as a really good Nocturne depends on the tactical decisions he can make on the enemy team.
I don't understand assassin Rengar very well (I usually only pay attention to tanky Rengar), so I can't help you there, but in general, past the rule of waiting for the enemy to burn their CC, you're going to want to evaluate what the Assassin you're playing is strong at and work with it.