Before we get into specific champions, there's one rule that applies to countering every assassin in HOK: assassins are cooldown-based. When their abilities are up, they're the most dangerous champions on the map. When their abilities are on cooldown, they're walking gold.
The entire game of playing against assassins is about forcing them to use their cooldowns on the wrong target or at the wrong time. If Ying uses her full combo on your tank, she has nothing left for your ADC. If Han Xin uses his dash to engage and you flash away, he can't follow up. If Zilong blows his ultimate and your team disengages, he just wasted his strongest cooldown.
This guide covers how to apply this rule for each role, plus specific champion counters.