Although there are a lot of KP farming methods for an early game, it's not really needed because you won't need too much KPs in an early stage. For those who still wants to, do the Frozen Continent gateway using Multichain + 3x KP. Even you enter with Lv100 you still almost 200 KP, by beating 9 enemies.
To begin with the REAL KP farming, just heading towards the very end of Scenario 000. It is strongly recommended (although other ways do exist) that you get yourself a lai strike buide. To get the Smiting Soul, fight card characters (e.g. Yuna) with high luck build. (Don't use low level multipier build! I wasted a lot of time on this and figured it out that you should simply stack up LCK, and then use a Lv100 character. It only needs 2-4 tries to battlegen Smiting Soul.) And you should have all multipliers you need now (purchasing them from Moogles). Buy 5 KP doubles and do The Forsaken Land gateway, you should be able to get 40000+ KP within 30 minutes.
The article below explains all the details needed for KP farming. Replace his suggestions with mine, which is his last suggestion.
Can be located in various chests on the world map, but more importantly can be bought from the Cornelia Moogle on either of the two days following your Bonus day. Alternate your PSP's clock between those two days and buy five of the skills, as he only sells one a day. You may need to complete 013 story or obtain 1500 KP (the exact trigger is unsure) to get the skill to appear.
EX force absorption > max
Wall Rush damage > max
Other settings are unimportant
Then equip your character with:
Force to Courage accessory (obtained from Accomplishment 069: Bonus Round, successfully complete AP chance 100 times)
Pearl Necklace (optional, but helps)
What this does is turn up the absorption rate for EX to the point where even a little bit of EX force would normally max your EX gauge. With Force to Courage equipped, this means that same little bit of EX force will instantly max your brave to 9999.
All you really need to do to win battles is use a single quick attack that causes chase. Go into chase, and use an HP attack. Your brave should max out instantly unless the computer refuses to dodge at all. All battles should look like this.
Attack > Chase > HP > Win
If the computer dodges your HP attack, reset and try again until it doesn't.
This can be combined with Iai Strike or Level 1 builds effectively.
And at late game, it is suggested to try in the "The Dragon King's Gateway" which has both a silver and gold Moogle Statue. The dungeon has 10 floors, and the KP chance is win in under 60 seconds.
Starting KP 99, x32, x2, x3 = 19008 KP minimum (even if you fail every KP chance)
If you complete every single KP chance, you get around 28000+ KP. With the above method, battles only last around 10 seconds average anyway, so that shouldn't be an issue.
The dungeon at the north eastern island "The Forsaken Lands", has 2 silver Moogle statues and 1 gold statue. Because the KP chance is to break in 60 seconds, use the Iai Strike method is recommended. You could get around 53000+ AP if completed all KP chances.