Flavour over Fluff
The key to understanding 3.5e/PF classes is that while they have evocative names (Rogue, Fighter, Assassin, Archmage) and a big pile of writing at the front that says it tells you what they do, often the mechanical effects of the class are not in line with the fluff - and another class might make a better version of the concept you are going for than one with a name closer to the concept.
For example, if a player came to me saying they wanted a character with innate magic they didn't know how to use gifted by dragon blood, and were struggling to control, i'd say Warlock without a second's thought. The Sorcerer class is word-for-word the fluff of what they are describing - but the mechanical abilities are extremely difficult to theme as 'innate, not-controlled magic' - it takes a skilled roleplayer to do so, and one who knows their way around the spell list. Warlock, though, is blasting people with raw magical energy and a few tricks that themselves are weird and not entirely what you might want. It's almost the ideal 'guy with magic who has no idea what he is doing' class, if you ignore the fluff entirely.
Fluff is the writing on a class that has no mechanical effect. As only the mechanics of a class actually translate into the game - your character's story and explanation for his powers supercedes what is written on the class - if you are concerned with flavour over mechanics, oddly, the first thing you have to do is look at the mechanics of any class you are thinking of applying to your character. As those are the things you can't change.
While many people are fine to use the existing fluff of a class, people heavily concerned with flavour will often want to change things up, refluff, or rewrite it to suit their specific character. As the only things you can't change are mechanics, those are the parts to pay attention to - because they are the ones that can stab your flavour in the back.
How To Make An Assassin In PF
Ninja. Ninja has a very eastern feel to it's writeup, but if you look at the abilities it grants, you quickly see there's nothing eastern about the effects. A shadow warrior concerned with stealth and striking swiftly from darkness, the ninja is much more like an assassin than the rogue - but a specific kind of assassin, not one who strikes from every direction and works the social aspect, but one who relies on physical and magical ability to climb up the side of an impassable castle and kill a king in his bed before disappearing into shadows and smoke just as the palace guard burst in.
Rogue. Rogues have the option to pick up some ninja abilities, as ninjas do rogues, but the cross-pollination is not as heavy as you might think, as the iconic (and defining) mechanical effects of the class remain class-specific. Rogues have the option to be sneakier, and dirtier than ninjas, which gives them more power due to the ability to have an indirect, or 'Face' like approach, but gives them slightly less raw 'teleporting through shadows murdering people'.
Vivisectionist Alchemist. This is the most assassin-like base class. You kill people, have a deep and enduring knowledge of anatomy, and use poisons. You also make magical potions that you use to empower yourself to kill people. It's the closest to famous fantasy-assassins, from the Witcher to Durzo Blint to FitzChivalry or Chade. Knowledge of secret herbs in fantasy = alchemy, and it and poison are hand in hand.
Mixing or matching these classes with a splash in some others could work, notably Fighter, Swashbuckler, Barbarian, and to a lesser extent, Inquisitor/Magus, both to represent a higher level of martial ability, or add more 'tricks' up your sleeve. Note that Magus/Inquisitor add martial ability, as does fighter - swash and mostly barbarian add tricks.
Overall, those three base classes are what you are going to be spending levels in to be a fantasy assassin. Vivisectionist gives you the most bang for your buck, but the other two are also fine if you know what you're getting into and dance past the trap options built into both classes.
Theoretically, a monk could be an assassin, with a decent level of monk mixed with rogue, and relying on Stunning Fist and feats that change/boost that ability, but that's a very specific flavour.
Assassin. Assassin's a bit of a poser, because it does it's very best to straitjacket you into acting a certain way as hard as it can. But mechanically, it does only a very few things for you. Sneak attack progression, death attack, true death, and hide in plain sight. The first is self-explanatory, and most 'sneaky' prestige classes are immediately noped out of because they don't grant it. Death Attack is.. interesting. We'll get back to it. True Death is completely boring - so many low level spells, even alchemist's fire, do similar things. There are mechanisms for this not tied to 4 levels in a prestige class. Hide In Plain Sight is great, and fits your fluff perfectly - sadly, it requires level 8, and Assassin requires level 5 to get into - so you're getting this at level 13, aka, well into the game if not by the time the game is over.
Death Attack is really what we come back to. There are optimization resources for magic items, feats, and abilities that increase the DC - most of them are 3.5e. The existing DC will not be very high. Even if you pimp out your Int, and enter Assassin as soon as possible, no monster will die to it except caster-types - and no fighter-type npc will die to it, unless they are not wearing rings of resistance or the like at all. It's a huge let-down when you try this, and it doesn't work, so if you want it, you'll need to really invest in it, not just your character's resources, but also your time in researching how to improve it. It's a weak ability, so it's no small potatoes to try to improve it. On the gripping hand - poisoned dagger full attack with sneak attack from the shadows will likely kill anything that would die to that DC in the first place. So that's a question you have to ask yourself. What is this actually giving me?
Shadowdancer. This class is important because, while it requires 3 crappy feats to enter, it grants Hide In Plain Sight at first level. If it gave sneak attack progression (it doesn't), it would be a shoe-in. The abilities are good, and suit an assassin, especially once you strip the 'agent of the darknesses!' fluff from them. But it doesn't add to your murder ability, and that will start to hurt you as you level up. And the good abilities don't come thick and fast enough to make up for that. A 1 level dip costs you 3 feats and 1 level - but it lets you hide anywhere, and that is invaluable. This class + Assassin at the same time would be the ideal assassin prestige class.
I looked at all of the other prestige classes that seemed 'sneaky' but they were, to put it mildly, crap. Boring mechanics, and boring fluff. I really don't see any that even look slightly good.
I see three major options for you, to make an assassin in pf (as opposed to just a sneaky person) -
Vivisectionist Alchemist 5/Shadowdancer 1/Vivisectionist Alchemist 14
This gives you your magical poisoner/assassin, ala Durzo Blint or Chade. You can dip part of your later Alchemist progression into Assassin to pick up Death Attack, but it's that question again. Alchemist makes you progressively stronger as you get up in levels, enough to keep pace with casters. Rogue 3 gives you Improved Uncanny Dodge and Improved Evasion when combined with Shadowdancer 2, but probably isn't worth the dip.
Rogue 5/Shadowdancer 1/Rogue 14
This gives your silver-tongued, dark-eyed social assassin, who enters the enemy's bedchamber disguised as a maid. The lies and intrigue make you somewhat like the old 'Monster' field agent concept, with a soul as black as tarnished silver. You don't need to dip Assassin with this build, as several Rogue advanced talents are similar to Death Attack, the 'knockout blow' and 'deadly cocktail' being the most obvious. You can actually strip shadowdancer out of this if you want to spend the feats elsewhere - you rely less heavily on stealth than the other two builds, and can pick up Hide In Plain Sight from an advanced talent.
Ninja 5/Shadowdancer 1/Ninja 14
These builds all look pretty similar. And that's because in PF, very few prestige classes are worth it. Ninja has an advanced 'ninja trick' called 'Assassination' that uses word for word the same text as Death Attack. They also get many things that help get into position to assassinate, that the assassin doesn't get.
This is actually the closest to a knife in the darkness, a mundane assassin, that you'll find.