You want to pick Kai'sa when paired with an engage support and if you're not against poke in the botlane. Poke is kai'sa's worst enemy as she is extremely short ranged and will get poked out of lane. This is why you don't want to blind Kai'sa as she gets hard countered by a lot of poke supports.

The early game as Kai'sa really depends on the matchup and your support.
If you lost the dice roll and got an enchanter then brace yourself to play on the defensive. You can still look for trades just play around your enchanters cooldowns and trade accordingly.

IF you're fortunate enough to play with an engage support then brace yourself for a skirmish heavy lane. And in these skirmishes it's really important to come out on top.

That's why you need to know how to trade properly.

Generally with Kai'sa your bread and butter trading pattern is auto Q. Auto Q is really good especially with taste of blood as it gives you sustain while being a good burst trade. It's important to note that if you go for this trade then you want to instantly retreat as soon as you do the combo when you're trading in a minion wave. After you auto Q you're generally out of gas and need to wait until your Q is back up. And if you keep fighting after Auto Q you run the risk of the enemy trading back and this will generally lead to a lost trade.

Another combo you can do that is especially good with engage support is to w them while they're cc'ed. This will almost always lead to a passive proc and even if it doesn't you will still nearly always win the trade and you should as a rule of thumb always hold your w until you can burst with your support. W'ing randomly in lane is unreliable and will get punished as your burst is far worse when it's on cooldown.

Freezing as Kai'sa can work but counters both your W and Q. You can go for it in certain situations but generally I like to hold the wave in the middle. This leads to good engage options while there is less minions to tank your Q and block your W. This also makes it easier for your support to land skillshots as they can't hide behind as many minions and engaging will be easier.
Note that freezing is a viable strategy that can lead to kills post six as your engage support if they don't have skillshot cc can easily engage when the enemy overextends for cs.

Mid game as Kai'sa is where she thrives the most with the current build and meta. This is where you want to capitalize on your one shot capabilities and make plays on the map. Get vision down and utilize your blue trinket and W to find and land picks. You generally want to go mid as this is where you get the most access to the map and have the easiest time rotation. Fighting in the mid game is really beneficial for you as you can easily join fights and reposition with R and E. Play the mid game as if you're half assassin but you're more reliant on your team to setup.

Late game is where you really want to group and stand behind your allies. Depending on the comp you either play to dive their backline or play front to back. But always be on the lookout for prime opportunities to get a pick or jump into their backline to assassinate their backline. But be careful. One bad ult can lose the game so don't just jump carelessly into their team. Play smart and calculated and you should come out on top.

If you're against hard poke champs as Xerath you generally want to look to jump him. Playing front to back really sucks against poke as you get chunked out of the fight and have to retreat. So do your best to blow them up before they make you irrelevant.