What does (or should) an Assassin-class character do better than anyone else? What should its niches be? What should its weaknesses be? Let's start from there, basing it off Rogue.
What does (or should) an Assassin do better than anyone else, and what should its niches be?
Disguise. This should be a true niche of the class; have them be by far the best at physical disguises (which, note, things like
True Sight will still be fooled by); other classes can try this at low-ish odds of success but Assassins can do it almost perfectly right out of the gate.
Poison. This should also be a class niche. Assassins are trained to recognize poisons, apply and administer them, treat or cure them with antidotes, and - at later levels - develop their own. Other classes can try some of these but at low-ish odds; except apply and administer which anyone can do only the Assassins is at much less (or zero) risk of accidentally administering poison to him/herself.
Killing. Against the right sort of targets (I'd suggest humanoids only, with size range increasing as level advances) this should be a class niche: only Assassins can one-shot kill on a good strike. This would require some tweaking of how 5e handles damage; in the specific case of Assassins a melee sneak attack that rolls 5 or more over what's required to hit bypasses hit points completely and either a) kills the target on the spot if the target is equal or lower level/HD than the Assassin or b) puts the target at 0 h.p. and into death saves if above the Assassin's level/HD. On a roll to hit that succeeds by less than 5 normal sneak attack damage still applies. Best of all, when attacking an unaware opponent they'd strike as if a Fighter of their level for the first attack.
Higher-level Assassins can use two weapons when doing this, with either one capable of a one-shot kill.
Stealth. Shared niche with other Rogue-type classes, with abilities about on par with Rogue for things like hiding, stealth, etc.
What should its weaknesses be?
Frontal melee. They're just not very good at it. They're unable to use anything other than light armour and their combat training (other than for killing strikes) is only slightly better than a Wizard.
Alignment. They have to be evil, because - and this is a 1e definition that I quite like - poison use is an inherently evil act.
Killing. Yes, this is also a weakness: the Assassin only gets to attempt the killing strike attack
once per encounter; and if it misses the opponent(s) become(s) aware of the Assassin and the Assassin's in trouble. Note this differs from Rogue sneak attacks which they can keep using; an Assassin only gets one good shot.
That's enough to start with...
