Paizo is really good at setting and story, so start by just using it as is from the Pathfinder books. Likewise their default setting of Golarion is very richly detailed, and so if it is important to the adventure, and you're not somewhere else already, then set it in Golarion, otherwise set it somewhere undefined. Your players only need to know enough of the world to know where they have come from and where they are going.
However, some aspects of the fluff of the setting have impact on the crunch of the game. Deitys can provide mechanical benefit to clerics and other divine classes; magic items can be tied strongly to the setting; and so on. A theme I'll come back to over and over again, is to try to do as little as possible, and to try to just hand-wave as much as you can.
If fluff elements are not critical to the adventure or adventure path, it's probably easiest to just swap the adventures fluff elements out for the equivalent elements from 4e.
- if it's not relevant to the adventure that the legendary axe of a specific dwarfish lord of legend has a very particular magic power, then make it a magic axe that already exists in 4e.
On the other hand if the fluff elements are important to the plot, then you'll need to convert them, sometime that can be easy - it may not matter too much that Pharasma (the Golarion goddess of death) is not associated with winter, while the Raven Queen (the default 4e goddess of death) is, and if that difference isn't going to be weird, then accept it. A 4e cleric of Pharasma can take Raven Queen associated channel divinity powers, and perhaps even Winter domain channel divinity powers. If you're using the WotC character builder, that character is a (mechanically) a cleric of the Raven Queen. But in game a cleric of Pharasma.
But if all else fails and you need customizations, if your players use the character builder, the most difficult areas will be ones that have mechanical effects not found previously in the game.
A foot slot magic item that gives bonuses to all your defences is going to hard for your players to mimic in the builder.
A cloak that gives a bonus to all your non AC defences as well as granting a unique power is much easier for them - they can take a basic +X necks lot item with no powers - making the numbers on the character sheet correct, and then write in the special power.
Don't try to convert stat blocks and other similar crunch.
4e monsters are super easy to reskin. Kobold Wyrmpriest seem like the right monster but you need a goblin? Replace shifty with goblin tactics and call it done.
NPC's in 4e use monster statblocks. If you want to make them feel like members of a class, given them an iconic at will from the class (and maybe an encounter too) - there is no need to spend time trying to create a "correct" build of the character.
CR is used in Pathfinder in the same way that Level is used in 4e - Though 4e also adds minion, elite and solo designations. While character levels in Pathfinder fit into a 1-20 range vs 1-30 in 4e, the CR range (and CR in theory is analogous to level) runs up to 39 which is higher than 4e's toughest monsters. I would just treat pathfinder CR and 4e level as almost equivalent.
4e assumes that you are not fighting monsters far lower in level or far higher in level then yourself, as the low level side will almost never be able to hit the high level side.
Essentially, I would boil this part of the question down to "convert the monsters as described in Best way to convert creatures from 3.x to 4E?" and then try to keep as much of the rest as is.
Pathfinder's skill list is more detailed than 4e's, but for the most part they can be easily mapped onto their 4e versions. Climb is covered by Athletics, and so on.
Skills like Handle Animal, Use Magical Device and Profession cause more problems, and probably need to be dealt with on a case by case basis (and probably at a call for use level).
In non stressful situations, Handle Animal and Profession can probably be either an automatic success if a player has a good in character excuse (background, theme, class, or whatever) for knowing how to do it, or a die roll at a reasonable stat + 1/2 level if they are untrained.
- In a difficult or stressful situation, if the player can provide that reason for knowing the skill, Profession could just be a reasonable stat + 5 + 1/2 level just like any other skill or impossible if untrained.
Use Magic Device poses different problems as it exposes a game mechanic that is fundamentally different between 4e and pathfinder. I would try to ignore the need for this skill and it became important use Arcana as a substitute.