Assassins have the ability to just pop in, melt hp and leave. Which creates a terrifying presence when you appear in team fights. As long as an assassin doesn't get targeted, cc'd, or kept in the back line, it can do amazing amounts of burst damage, while being able to escape with enough life to run to wraiths, heal up with lifesteal and go shred another target. Hard hitting bursts, mobility to free roam wherever you want, terror in the enemy once you start snowballing.
If the enemy team is more teamfight friendly, if you don't have someone to disengage or stop their engagement then it might become a problem. With someone like sona, use her ult to stop their engagement, and then the assassins dive in, catching them offguard and shred them apart.
The most weak thing you have to watch out for is if your assassin carry gets caught out of place. Similar to the old meta, if the adc got caught and died, the enemy team would push hard feeling no threat was alive.
Lha'zix new changes makes him a great dualist, so a 1v1 would be easy for him. But taking on a team like his old assassin ways just doesn't do it as well as a zed or talon. With the new MR items, AP assassins like leblanc, make it hard to melt as much as they used to. Although with rush of DG, it kind of counters heavy MR builds slightly. I still think leblanc would be nice, but she isn't as assassin-like as other choices.
For a good assassin mid lane, I find wave clear with mobility to do single targetted or aoe damage works best. Like Kat's ult, or Akali's dives. Or Zed's everything. If you can do a lot of damage without getting close to them, and still have a decent amount of mana/energy/whathaveyou, you should be alright. You just want to be a bully to them, and out farm them. Quickly gain that exp and gold advantage, get them scared and catch one of their mistakes and capitalize one it.
If you think about old meta vs new, you used to have ap carries like brand or annie in mid lane, who sit and farm really well and roam when they get that tower down to try to bully other lanes while the enemy mid champ is recalled or something. If you try to run that against an assassin like zed or kat, you might find yourself getting bullied out of lane fast. To counter it you either play champs like fizz and ahri, or ad assassins like talon or zed or kassadin. To be able to do it right back to them, or have someone like orianna who just gets to stay safe away from assassin dives and can farm with her q, w combo.
Sorry if I ended up repeating a lot, or if some parts get confusing. This is just my input on it all.
