I've mained as Kai'Sa for quite a while now. These are several strange experiments mostly resulting in positives outcomes.
#1 thing to remember as Kai'Sa: the possibilities are endless.
#2 thing to remember as Kai'Sa: except tanking. Don't try it. Please.

This Guide is not very indepth and is simply meant to provide special builds. This guide will cover choice of summoner spells and rune builds.

Note: The "Not-so-nice Kai'Sa's" are NOT recommended for PvP play and are simply meant to show what NOT to do. You can still mess around with them on PvE if you want to. I know I do.

Hail of Blades: Don't underestimate a strange rune. Hail of Blades is a cheeky method of rapidly applying stacks of plasma onto a target. When combined with Void Seeker, Kai'Sa can apply 5 stacks and detonate Second Skin on a target champion within less 3s, even at level 3.

Sudden Impact: Because this rune is activated after Supercharge and Killer Instinct, Supercharge becomes the central skill for this build. Killer Instinct is particularly useful against melee champions, as Kai'Sa can exploit it for it's shield and lethality through Sudden Impact.

Eyeball Collection: Kai'Sa's other potential choice in this tree is Zombie Ward, however I find the extra AD provided by Eyeball Collection to prove more helpful, simply because Kai'Sa doesn't act alone very often and can thus rely on allied champions and her own Stealth Ward for ward vision.

Relentless Hunter: The only other option in this tree would be Ravenous Hunter, which offers slightly better survivability. However, because most of AD Kai'Sa's damage comes from Basic Attacks rather than her abilities, it's more worthwhile to invest in greater out-of-combat MS. Relentless Hunter has saved me a more than a few times. Legend: Alacrity: Kai'Sa loves the extra AS. The faster she can apply stacks, the faster she can detonate Second Skin. Reminder: minions DO CONTRIBUTE towards Legend stacks.

Coup de Grace: More confident players may prefer Triumph for its bonus gold and health restoration, but Coup de Grace allows Kai'Sa to finish off champions more easily and force them out of her lane, granting her the opportunity to gather gold and exp with ease. This can prove especially useful early game during the race to level 2.
Flash: Although other champions may have more fortune with Flash offensively, all champions love its escape potential, and Kai'Sa is no exception.
Ghost: Much like Flash, Ghost has both offensive and defensive capabilities, but with a lower cooldown ( Flash=300s Ghost=180s). It's an unusual choice these days, but it can work just as good as Heal or Flash if utilized properly.

Lethal Tempo: Any bonus AS is nice for applying plasma stacks. Lethal Tempo can take a bit getting used to, but it can be more effective than Hail of Blades if used correctly. Lethal Tempo is triggered by Void Seeker, so Kai'Sa can strike with Void Seeker from afar before closing the distance with Supercharge and profiting from the bonus AS.

Triumph: The other runes in this tree are not options. Triumph far exceeds the other runes in terms of benefits. The bonus gold alone is enough for me to choose it.

Legend: Alacrity: Kai'Sa loves the extra AS. The faster she can apply stacks, the faster she can detonate Second Skin. Reminder: minions DO CONTRIBUTE towards Legend stacks.

Coup de Grace: Last Stand may be a tempting option, however Last Stand works better with sturdier champions who can spend more time at low health. Coup de Grace works great for finishing off a champion with Void Seeker. Sudden Impact: Because this rune is activated after Supercharge and Killer Instinct, Supercharge becomes the central skill for this build. Killer Instinct is particularly useful against melee champions, as Kai'Sa can exploit it for it's shield and lethality through Sudden Impact. I recommend using Void Seeker to trigger Lethal Tempo rather than with a Basic Attack followed by Supercharge so you can make full use of the 5s of Lethality.

Relentless Hunter: The alternative to this is Eyeball Collection, but I find the extra MS to prove more useful for escaping or dashing to saucy situations. Ravenous Hunter is, as I've discussed before, weaker in an AD build due to AD Kai'Sa's preference for Basic Attacks.
Flash: Although other champions may have more fortune with Flash offensively, all champions love its escape potential, and Kai'Sa is no exception.
Barrier: When Lethal Tempo is charging and enemy champions start to target Kai'Sa, Barrier can prevent some damage from making contact with her HP and allow her to begin her attack with the now-boosted AS. The cooldown for Barrier is tolerable as well (180s).

Press the Attack: Press the Attack has limited early game potential compared to the other main Precision runes, but makes up for it with great late game potential. Press the Attack also increases the amount of damage dealt when Second Skin detonates on a champion.

Triumph:The other runes in this tree are not options. Triumph far exceeds the other runes in terms of benefits. The bonus gold alone is enough for me to choose it.

Legend: Alacrity: Kai'Sa loves the extra AS. The faster she can apply stacks, the faster she can detonate Second Skin. Reminder: minions DO CONTRIBUTE towards Legend stacks.

Coup de Grace: Last Stand may be a tempting option, however Last Stand works better with sturdier champions. Coup de Grace works for killing a champion before they have time to react. Celerity: The second best choice would be Transcendence, however an AD Kai'Sa build isn't as dependent on abilities as she is MS. Most items that provide crit chance and AS also provide MS, creating good synergy with Celerity.

Gathering Storm: This rune is excellent for maintaining Kai'Sa's power late game, as AD Kai'Sa lacks the raw lategame power that AP Kai'Sa has. There isn't an alternative for Gathering Storm.
Flash: Although other champions may have more fortune with Flash offensively, all champions love its escape potential, and Kai'Sa is no exception.
Heal: This build doesn't benefit from Ghost or Barrier as much as it does from Heal. Heal provides the early game sustain Kai'Sa needs as well as some movement speed for evading enemy champions. Its cooldown is a bit... lengthy, but it works as a really nice escape spell.

Phase Rush: Depending on how AP Kai'Sa is built and your play style, you may go for either Arcane Comet or Phase Rush. Arcane Comet is better if you prefer to attack from afar with Void Seeker, but Phase Rush is better if you prefer to attack at close range. Phase Rush is helpful for sticking to a champion that's retreating so you can detonate Second Skin or for displacing to a safer position. Phase Rush is NOT triggered by a single Icathian Rain, since Icathian Rain only counts as one ability attack regardless of how many missiles hit a champion. Using Icathian Rain against multiple champions also DOES NOT trigger Phase Rush. Each first missile that hits an enemy champion counts as a single attack against that champion. Phase Rush is triggered by three attacks against a single champion.

Manaflow Band: If you prefer to fight at long range, Manaflow Band is your best choice. The mana restore it provides greatly extends the time Kai'Sa can stay out. If you prefer close range combat, Nullifying Orb is the better rune. It will act as your magic defense, allowing Kai'Sa to avoid magic resist items in favour of more powerful AP items.

Celerity: AP Kai'Sa doesn't have as many movement items as AD Kai'Sa has available to her, so in order to avoid becoming too slow, Celerity is the best rune, particularly for short range Kai'Sa's. Celerity also provides more AP when Phase Rush is activated. Long range Kai'Sa's may instead use Transcendence so that they can fire Void Seeker faster.

Gathering Storm: AP Kai'Sa doesn't have enough early game fire power to make use of Scorch, so Gathering Storm is your best option. Might as well make late game Kai'Sa even more powerful. Legend: Alacrity: Kai'Sa loves the extra AS. The faster she can apply stacks, the faster she can detonate Second Skin. Reminder: minions DO CONTRIBUTE towards Legend stacks.

Coup de Grace: More confident players may prefer Triumph for its bonus gold and health restoration, but Coup de Grace allows Kai'Sa to finish off champions more easily and force them out of her lane, granting her the opportunity to gather gold and exp with ease. This can prove especially useful early game during the race to level 2.
Flash: Although other champions may have more fortune with Flash offensively, all champions love its escape potential, and Kai'Sa is no exception.
Ignite: Luckily, nobody knows that Cleanse removes the flame from ignite, so feel free to use ignite at your leisure. Ignite also reveals enemy champions, so you can use this to track your target to fire at with Void Seeker.

I can hear the meta players cheering for this rune Fleet Footwork: This rune scales heavily with AP, so using it in an AP build can extend its usefulness into late game. Fleet Footwork is a great for its healing and bonus temporary MS. Jungle Kai'Sa loves the extra sustain from it (I've jungled here and there, so I know how tough early game is on HP). The MS it provides is nice for both escape and chasing targets. However, Fleet Footwork loses its effectiveness late game, as opposed to Press the Attack, which remains effective the entire game. The easy trick to may full use of this rune is to kite. Attack, then move, then attack, then move, etc. This trick becomes annoying late game, but it's tolerable before then.

Triumph: The other runes in this tree are not options. Triumph far exceeds the other runes in terms of benefits. The bonus gold alone is enough for me to choose it.

Legend: Alacrity: Kai'Sa loves the extra AS. The faster she can apply stacks, the faster she can detonate Second Skin. Reminder: minions DO CONTRIBUTE towards Legend stacks.

Coup de Grace: Last Stand may be a tempting option, however Last Stand works better with sturdier champions. Coup de Grace works for killing a champion before they have time to react. Jungle Kai'Sa loves this rune for stealing kills from the bushes. Celerity: The extra MS is fantastic for Jungle Kai'Sa, due to her roaming nature. I wouldn't recommend using Transcendence.

Gathering Storm: AP Kai'Sa doesn't have enough early game fire power to make use of Scorch, so Gathering Storm is your best option. Might as well make late game Kai'Sa even more powerful. If you're really confident, you can try Waterwalking instead if you want to target their jungler and the drake more than their laners, though that will limit Kai'Sa's late game potential.
Flash:Although other champions may have more fortune with Flash offensively, all champions love its escape potential, and Kai'Sa is no exception.
Smite: Smite unlocks the jungle items, which give bonus exp against monsters. I prefer to upgrade it to Challenging Smite for it's damage reduction, but you can go for Chilling Smite. Both are nice spells.

These are some rune builds I've had some success with. You obviously don't have to follow one exactly. Everyone of them can be modified (I'm still experimenting with Lifesteal Kai'Sa, with... limited... success). Inspiration can be used as a secondary rune, but I don't recommend it. Stay away from Resolve runes.

Feel free to express your thoughts.