Maybe I'm missing something, but I can't see any reason to ever take this feat. The only time it provides a bonus is if you NEED to long/high jump to get to your target for some reason. Otherwise, it has the exact same action economy as Stride+Strike or Leap+Strike.
It seems to unlikely for this to even come up for the feat to be useful. I've encountered situations like this maybe once or twice in seven years of tabletop, and it seems like it would always be better to just take a slight hit by using an extra action than sacrificing a feat to save one action in a campaign.
It is competing with decidedly more useful options, like Wholeness of Body, Dancing Leaf, Deflect Arrows, Wolf/Tiger/Dragon/Crane Stance, Ki Strike, etc. Any one of these is likely to be useful multiple times per fight or day.
I'd rather use Stunning Fist, even though it kind of sucks this edition (only stuns on a crit? Why?). If it let you jump more than your Speed (since it's only a straight line), or deal bonus damage/apply conditions with the attack, I could see the value.
It saves actions if you're going to long jump (and you can stack a pretty big jumping modifier) since that's already 2 actions. Who knows how often that's likely to come up though.
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The comparison point is to Sudden Charge from Fighter 1. That's also a two-dot action that lets you Move and Strike, but Sudden Charge lets you move twice your normal speed. Flying Kick letting you Leap is absolutely a trap, because Leap is already a one-dot action and you'd much rather Leap and then Flurry than Leap and then Generic Strike; even if you would use a generic strike after the movement there's no benefit to it compared to just using the actions normally.
Flavor aside, I feel like Sudden Charge would be stronger for Monk than this is. It'd take advantage of the Monk's crazy base speed and it doesn't require straight-line movement. Alternatively, give the Strike after landing some kind of bonus such as making the target flat-footed against the attack.
I brought it up as an aside at the end of my http://paizo.com/threads/rzs2vad4?High-Jump-So-epic-so-we-made-it-high thread. I didn't want to make another post just for it, but I'll happily add to this one.
Flying Kick is pretty easy to make basically useless, which is really sad for a level 4 class feat. If you're just going to Leap, it does nothing (leap is already one action, you can strike when you land for a second action). Maybe there's an edge case where you don't provoke a reaction or something, I dunno.
If you take the "Quick jump" skill feat (a level 1 skill feat requiring only trained in athletics), Flying Kick does basically nothing for action economy if you just need to jump. Quick jump makes high jump and long jump only a single action that does not require the additional movement at the beginning. Quick jump isn't a great feat because the DCs on those jumps are really high, but that's a different topic.
Okay, so what's the best case scenario for flying kick? Technically the base long jump requires moving 10 feet first, so if you want to move at least 10 feet before you do a potentially longer leap that you are likely to critically fail for your remaining movement, I guess that's an option long jump (and thus flying kick) provides.
Level 1: Nimble ancestry feat (+5 speed), Crane Stance monk feat (+4 to jump), Trained in athletics
Level 3: Expert in athletics, Fleet general feat (+5 speed)
Level 4: Dancing Leaf monk feat (+5 feet to Leap or successful high/long jumps), Powerful leap (increase distance of horizontal leap by 5 feet)
Level 6: Incredible movement (+15 speed) Flying kick monk feat, I guess, since that's what we're looking at
Level 7: Boots of Bounding (level 7 item, gives 5' more horizontal movement on Leap and +2 to high/long jump checks, increases speed by 5)
Total:
Speed: 60 (30 elf, +5 untyped (nimble), +5 untyped (fleet), +15 conditional (monk), +5 item)
Athletics bonus on long jump: +19 (+4 Str, +1 Expert, +7 level, +4 crane stance, +2 item)
For reference:
attempting to move during your Leap (so Leaping 20 feet would require a DC 25 check). You can’t Leap farther than your Speed. This DC might be increased or decreased due to the situation, as determined by the GM.
Success You Leap, increasing the distance you move horizontally to the desired distance.
Failure You Leap the maximum distance allowed by the Leap action.
Critical Failure You Leap the maximum distance allowed by the Leap action, then fall prone.
So we have a 50% chance of getting an 11 or higher, which with our +19 means hitting 30. That's a 25 foot leap in a long jump. Except that we have a few feats that modify that, so let's stack them up. Powerful leap boosts horizontal jumps by 5' unconditionally. Dancing leaf boosts a Leap or successful Long Jump by 5', except that a long jump includes an attempt to Leap, so, I guess it only counts as a leap if it's successful, except both failure and critical failure still says you Leap? Weird. The boots say they work when you use the Leap action, so we'll assume they don't help here.
Technically it says the DC is 5+ the total distance in feat you're attempting to move during your leap, so arguably all these feats do nothing, but I prefer to think of it as saying we want to attempt to jump 10' fewer than we actually do, and then add the 10' at the end because that's the best order to do it.
With the 10' boosts, it's:
DC 20 (only fail on a natural 1, but failure is the same distance anyway): 25 feet
DC 25 (75% success, 5% crit fail and fall prone): 30 feet
DC 30 (50% success, 5% crit fail and fall prone): 35 feet
DC 35 (25% success, 30% crit fail and fall prone): 40 feet.
You'd probably be better off always going for a DC low enough that a natural 1 isn't a critical failure. If you absolutely only had two actions to get there and attack, you might justify rolling for the extra distance on a leap.
So with a flying kick, you could stride 60 feet, guarantee a leap of an extra 25 feet, and get an attack, all in two actions, so you could follow it with a flurry (or start it with going into crane style, I guess).
Without optimizing for speed and jump distance with all the other feats and whatnot, it's not too hard to make the long jump a 45' stride (elf+monk) with a 15' leap at a DC that's only a failure on a natural 1 (which won't happen at level 7 without at least the item or crane strike). At that point it's basically just a stride+15'+attack in two actions, and the leap is nothing but flavor and a roll that lets you fail.
So it's still pretty bad, given the other options you could take. Maybe in a "maximum mobility at all costs" build it would find a niche role in that extra few feet, but it seems like it would otherwise be pretty situational. Maybe they were trying to keep it from being as powerful as sudden strike given the monk's overall mobility and the ability to flurry immediately afterward?