The assassin class was one of the innovations in Supplement II: Blackmoor, and hails more or less (there is some difficulty with who wrote what in the supplement; its editor, Tim Kask, has said that the assassin material was revised) from Dave Arneson's work. There has been ample objection to this class over the years, and it was used in 1e AD&D but excluded from 2e with the concept that an assassin is not a true "class" function. By that measure, a thief shouldn't be either; neither stealing nor killing is a pure class. Similarly, it is pointed out that assassins aren't heroic; again the thief speaks against ideals of pure heroism. Both are breaking major moral and ethical rules as a way to make a living. So none of the arguments against them really wash for me.
The assassin character type adds to the game a built-in structure, the Assassins Guild. This lasted into 1e, but was made optional for PC assassins; in Blackmoor they are all guild members, PCs included. It also makes the fascinating observation that assassins are neutral, whereas AD&D stuck them firmly in the "Evil" side of its alignment matrix. That certainly colors your Blackmoorish neutrality and chaos differently, I think, considering that thieves can be neutral or chaotic, and hired by lawfuls. It also works in terms of insulating assassins from either "side" when viewing law vs chaos as a fight with two sides instead of a grand ethical dilemma. Assassins are for hire and don't take sides.
An assassin's guild was part of Fritz Leiber's Lankhmar, built into the "Slayer's Brotherhood." This certainly seems much closer to the concept of the Blackmoor assassin than the somewhat muddled accounts of Islamic killers in various medieval sources. It has little basis in reality, but presents a fantasy setting with a very distinctly grey sense of morals. The mere existence of an assassin's guild means that society, tacitly or openly, accepts its existence.
Blackmoor introduced levels that require you to fight a superior, both for monks and assassins. With monks fights aren't necessarily to the death; with assassins they are. A Guildmaster assassin got there by killing his predecessor, period. AD&D added to this the Grandfather of Assassins, a level above that again can only be reached by more murdering. Too many assassins reaching higher levels automatically spells trouble, and a whole high-level campaign could be waged simply around a PC as a Guildmaster Assassin.
What is really interesting to me is the question of the assassin in a dungeon crawl campaign. Having an assassin PC seems like an invitation to basically switch over to a city-based campaign. But assassins are low-functioning thieves (2 levels lower than the already paltry thief numbers) per Blackmoor, and will be skilled poisoners as well. Consider - instead of the traditional "hack through the opponents" mission, an assassin PC could really thrive in a "faction dungeon," getting hired (with the available treasure) to take out assassinations on all the leader-types of different monster factions. It really puts the normal dungeon crawl on its head, but I think it's a fascinating possibility, even for solo play.
A final tidbit that I think wasn't followed up well enough appears in the Blackmoor assassin write-up: "Details of poison types will be handled in some future supplement when alchemists are fully covered." There's a bit more in the Dungeon Masters Guide, but doesn't really give us that full treatment I would like to see.