The difference between an assassin and a serial killer: paychecks.
The difference between an assassin and a soldier: politics.
The difference between an assassin and a hero: public opinion.
The difference between an assassin and a rogue: ... I honestly have no idea.
In truth, when you look it that way, assassin would be better as Background.
Assassin
Skill Proficiencies: Deception, Stealth
Tool Proficiencies: Poisoner's kit, disguise kit
Equipment A vial of basic poison, a set of dark common clothes including a hood, and a belt pouch containing 15 gp
Feature: Infiltrator
You can unfailingly create false identities for yourself. You must spend seven days and 25 gp to establish the history, profession, and affiliations for an identity. You can't establish an identity that belongs to someone else. For example, you might acquire appropriate clothing, letters of introduction, and official-looking certification to establish yourself as a member of a trading house from a remote city so you can insinuate yourself into the company of other wealthy merchants.
Thereafter, if you adopt the new identity as a disguise, other creatures believe you to be that person until given an obvious reason not to.
So with that, a Trickery Cleric, a Illusionist, a Thief, a Berserker or whatever could be assassins.