Setting
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.
Monsters
Don't try to convert stat blocks and other similar crunch.
Remember that:
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.
Skills
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.
Again, there's no easy way to do that.
You might be familiar with this assumption and I can tell you where it steams off too. D&D 4th edition is quite a different game with some similarities that were kept to brand it as "D&D".
The main problem here is that monsters only have powers they can actually use during an encounter. No more 1st level spells on your epic dragon. No more "the monster has this spells provided it has the time to buff". Everything in 4e happens during combat. And 4e Monsters don't have to deal with save-or-die spells, while most solos have to deal with multiple stuns (which didn't happen in 3.x).
AoO work in a different way, ranged attacks deal damage based on Dex, spells deal bonus damage based not on the level but on the mental scores. 5-feet step is now a movement action (except for kobolds)... Different game.
The easiest thing you can do is to take existing D&D 3.5 monsters or NPCs and present them to the group as if they were the kind of interesting monsters you want to portray.
Maybe choose something that has similar mechanics if you can find it.
In other word(s), reflavor.
Best Answer
The general principle of conversion is to keep the general sense of the original, but be prepared to throw out all the numbers and game mechanics which will not apply in the new edition.
Adventures
Many things can be used as-is. Dungeon maps, overland maps, plot hooks, general storylines, and so on, can be used with little or no changes.
When it comes to monsters, you basically need to create new encounters based on 5th edition monsters and encounter building guidelines, using the 4th edition encounters as inspiration. 4th edition encounters are too different: 5th edition has no equivalents of elites or minions, and its monsters may be different level to their 4e counterparts.
Give out treasure based on the D&D 5th edition expectations of gold and items, rather than the 4th edition standards. 4th edition had the assumption that you could buy items with gold, and that edition's treasure system was different in general. You probably want to insert similar items to the original where possible.
In general, you want to adhere to 5th edition sensibilities and design, forcing the 4e material to change to adapt to that, rather than the other way around.
Another thing to note is that 4th edition goes up to level 30, whereas 5th caps out at level 20. It's somewhat arbitrary how to translate the level difference, and you'll want to adapt the adventure to your party's level anyway.
Settings
Most of settings books can be used without modification. They're full of rules-neutral material like maps, history, politics and setting lore. All the basic concepts are still there, like fighters, wizards, characters who increase in level, and so on.
Some of the free 5th edition Unearthed Arcana material includes statistics for setting-specific material, such as the warforged and shifter races for Eberron.
Remember, also, that it's your world, and you're free to invent new things and fill in the blanks as you wish. A lot of world things don't actually require game statistics unless you're going to be fighting them.