Daughter of the Void

As a champion, Kai'Sa is an extremely flexible bot laner, who is playable with almost any support, and fits well into pretty much every team composition you can draft.
Though her nerfs in recent patches have left her feeling a little worse for wear, she still possesses one of the most versatile and dynamic kits in the entire bot lane roster. As such she is very much a viable pick, and with some practice you will be able to dominate games and 1v9 in a similar fashion to the likes of Vayne or Twitch at 4 items.

You will start Doran's Blade and one pot in 99% of your games. Q will be your first ability 100% of the time.
Level 1, you want to try and just last hit every minion with autos, using Q ONLY if you are being shoved in faster than you can manage with your autos. You can look for trades with autos level 1, but try to preserve your health pool for level 2.
At level 2 take W, this should be used very sparingly since it has a rather significant cooldown, and is vital for an all in early game, this holds true especially when playing with an engage support who can help you stack.
Level 3 you will take E, this is a fantastic tool for evading enemy engage, and retaliating with the attack speed steroid granted.
Max your Q after this, and play the lane sparingly, your primary goal is to maximise gold intake, not to force unnecessary fights, so focus on farming, and look to crash either the first cannon wave, or reset when you can crash the wave and have enough gold for either quiver or pickaxe. I don't reccmond backing at 1100 gold for dirk, since the quiver spike is stronger, and pickaxe would have been a more efficient use of time in lane vs gold intake.
If you find yourself obtaining early kills, I suggest the serrated dirk Q evolve rush, this is a very powerful trading tool, and will allow you to win fights with ease if you already have the edge over your opponents. If not, that is perfectly fine, and you can opt to either build collector anyway, or you can skip it entirely, and build kraken straight into phantom dancer, which will grant you your Q and e evolve, once your berserkers greaves have been built.
Your general game plan as Kai'Sa is to scale up to your 3 or 4 item power spike and take over the game, any kills you collect en route are a fantastic bonus, but do not panic if you aren't at the 'Epic Kai'Sa Pentakill Montage' stage by 15 minutes, play smart and you'll get there anyway.

TL;DR: Kraken > Galeforce in terms of consistency, and flexibility.

I get a lot of people asking me which item is better on Kai'sa. The short answer is Kraken, it is generally regarded as the better item in general since it works well vs all types of champions, whereas galeforce can sometimes leave you feeling a little lacklustre at times. That being said, both items ARE FINE and there is always an aspect of personal preference to deciding which mythic you want to build.

There are pros and cons to building both items, kraken slayer will begin outputting more damage than the galeforce active cloudburst after a couple of procs, so it is generally considered to be optimal for extended trades, and its effectiveness versus tanks and other armour stackers is evident, and should almost always be bought if there is no other source of high DPS on your team. Similarly, additional attack speed kraken grants as its mythic passive is incredibly useful once you start getting numerous items.

Galeforce will allow you to become even more slippery and evasive than you already were, combined with heal, flash, E and R you can ALMOST survive at full build vs a serrated dirk Kha'Zix. Galeforce is exceptional for helping you chase down enemies, winning short high damage fights, and finishing off low hp targets who either flash or heal etc away from you after you've spent R. Additionally, the dash is very helpful into comps where getting by skill shots will result in a swift death for you. It is worth noting, that galeforce pd will not give you e evolve as early as kraken due to its lesser attack speed stats.

Hopefully this clears up any confusion you may have had regarding mythic selection.

Kaisa has the ability to cheese opponents in a couple of ways.

The most common, and simple way, is using your abilities and autos in conjunction efficiently. For instance, aa R aa allows you to stack faster and turn the tide of a fight. aa W aa works in a similar manner, though this is more just correct auto weaving rather than an aa reset.

Other useful tips are things like, using e after q, so you have as little down time on dps as possible, though keep in mind during team fights than e is very important for evasive manoeuvres so use this sparingly.

Kai'Sa W has a couple of interesting interactions, casting while using R will launch the w as soon as you arrive at the blink location, this is useful for fights where you are cleaning up, or arriving late to the party, it's also great for finishing people off who still have flash as its very hard to react to the instant, point blank w.
Kai'sa can also use flash and w into two different ways. The first is simply flash w, which will fire your w from your flash location; the second is trickier to pull off and is sometimes referred to as a 'phantom w'. This is essentially where you flash towards the end of your w cast time and create a visual illusion of your w flying through minions or champions or whatever before hitting your desired target. In reality, the projectile fired from wherever you flashed to, but it looks as if you've just scripted or something, and can secure a surprising number of kills once you understand the trajectories a little more.